Question: What is another way to call the derivations of mathematical theorems?

Answer: formal proofs

<hl> Automated theorem proving refers to the development of computer programs that search and find derivations ( formal proofs ) <hl> of mathematical theorems. Finding derivations is a difficult task because the search space can be very large; an exhaustive search of every possible derivation is theoretically possible but computationally infeasible for many systems of interest in mathematics. Thus complicated heuristic functions are developed to attempt to find a derivation in less time than a blind search.

Automated theorem proving refers to the development of computer programs that search and find derivations (<hl> formal proofs <hl>) of mathematical theorems. Finding derivations is a difficult task because the search space can be very large; an exhaustive search of every possible derivation is theoretically possible but computationally infeasible for many systems of interest in mathematics. Thus complicated heuristic functions are developed to attempt to find a derivation in less time than a blind search.

Automated theorem proving refers to the development of computer programs that search and find derivations ( <hl> formal proofs <hl> )


Question: Why is sometimes finding derivations a difficult task?

Automated theorem proving refers to the development of computer programs that search and find derivations (formal proofs) of mathematical theorems. Finding derivations is a difficult task because the search space can be very large; an exhaustive search of every possible derivation is theoretically possible but computationally infeasible for many systems of interest in mathematics. Thus complicated heuristic functions are developed to attempt to find a derivation in less time than a blind search.

Answer: because the search space can be very large;

Automated theorem proving refers to the development of computer programs that search and find derivations (formal proofs) of mathematical theorems. <hl> Finding derivations is a difficult task because the search space can be very large; an exhaustive search of every possible derivation is theoretically possible but computationally infeasible for many systems of interest in mathematics. <hl> Thus complicated heuristic functions are developed to attempt to find a derivation in less time than a blind search.

Automated theorem proving refers to the development of computer programs that search and find derivations (formal proofs) of mathematical theorems. Finding derivations is a difficult task <hl> because the search space can be very large; <hl> an exhaustive search of every possible derivation is theoretically possible but computationally infeasible for many systems of interest in mathematics. Thus complicated heuristic functions are developed to attempt to find a derivation in less time than a blind search.


Question: What makes for a theoretically possible theorem be difficult to find derivations?

Answer: computationally infeasible

Automated theorem proving refers to the development of computer programs that search and find derivations (formal proofs) of mathematical theorems. <hl> Finding derivations is a difficult task because the search space can be very large; an exhaustive search of every possible derivation is theoretically possible but computationally infeasible for many systems of interest in mathematics. <hl> Thus complicated heuristic functions are developed to attempt to find a derivation in less time than a blind search.

Automated theorem proving refers to the development of computer programs that search and find derivations (formal proofs) of mathematical theorems. Finding derivations is a difficult task because the search space can be very large; an exhaustive search of every possible derivation is theoretically possible but <hl> computationally infeasible <hl> for many systems of interest in mathematics. Thus complicated heuristic functions are developed to attempt to find a derivation in less time than a blind search.


Question: What kind of functions are developed to find derivations quickly?

Answer: heuristic functions

Automated theorem proving refers to the development of computer programs that search and find derivations (formal proofs) of mathematical theorems. Finding derivations is a difficult task because the search space can be very large; an exhaustive search of every possible derivation is theoretically possible but computationally infeasible for many systems of interest in mathematics. <hl> Thus complicated heuristic functions are developed to attempt to find a derivation in less time than a blind search. <hl>

Automated theorem proving refers to the development of computer programs that search and find derivations (formal proofs) of mathematical theorems. Finding derivations is a difficult task because the search space can be very large; an exhaustive search of every possible derivation is theoretically possible but computationally infeasible for many systems of interest in mathematics. Thus complicated <hl> heuristic functions <hl> are developed to attempt to find a derivation in less time than a blind search.


Question: What proof verifier insists on  having a complete derivation as input?

Some proof verifiers, such as Metamath, insist on having a complete derivation as input.  Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants.  They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.

Answer: Metamath

<hl> Some proof verifiers, such as Metamath , insist on having a complete derivation as input. <hl> Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.

Some proof verifiers, such as <hl> Metamath <hl>, insist on having a complete derivation as input. Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.

Some proof verifiers, such as <hl> Metamath <hl> , insist on having a complete derivation as input.


Question: What proof verifiers can take a sketch and fill in the missing pieces?

Answer: Mizar and Isabelle

Some proof verifiers, such as Metamath, insist on having a complete derivation as input. <hl> Others, such as Mizar and Isabelle , take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". <hl> Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.

Some proof verifiers, such as Metamath, insist on having a complete derivation as input. Others, such as <hl> Mizar and Isabelle <hl>, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.

Others, such as <hl> Mizar and Isabelle <hl> , take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel".


Question: One way Mizar and Isabelle fill in the missing pieces is by?

Answer: doing simple proof searches

Some proof verifiers, such as Metamath, insist on having a complete derivation as input. <hl> Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". <hl> Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.

Some proof verifiers, such as Metamath, insist on having a complete derivation as input. Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by <hl> doing simple proof searches <hl> or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.


Question: How is the core that verifies the derivation called?

Answer: "kernel".

Some proof verifiers, such as Metamath, insist on having a complete derivation as input. <hl> Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. <hl> They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.

Some proof verifiers, such as Metamath, insist on having a complete derivation as input. Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core <hl> "kernel". <hl> Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.


Question: How are the results often formalized by these proof verifiers?

Answer: lemmas

Some proof verifiers, such as Metamath, insist on having a complete derivation as input. Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. <hl> Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas , for which derivations can be constructed separately. <hl>

Some proof verifiers, such as Metamath, insist on having a complete derivation as input. Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small, core "kernel". Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of <hl> lemmas <hl>, for which derivations can be constructed separately.

Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of <hl> lemmas <hl> , for which derivations can be constructed separately.


Question: What shows that if a first-order theory has an infinite model, then it has infinite models of every cardinality?

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain.  Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

Answer: The Löwenheim–Skolem theorem

<hl> The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. <hl> In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

<hl> The Löwenheim–Skolem theorem <hl> shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.


Question: Which extensions of first-order logic are more expressive and permit categorical axiomatizations of natural or real numbers?

Answer: infinitary logics and higher-order logics

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. <hl> Many extensions of first-order logic, including infinitary logics and higher-order logics , are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. <hl> This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including <hl> infinitary logics and higher-order logics <hl>, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

Many extensions of first-order logic, including <hl> infinitary logics and higher-order logics <hl> , are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers.


Question: First-order theories with infinite models can not be what?

Answer: categorical

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. <hl> In particular, no first-order theory with an infinite model can be categorical . <hl> Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be <hl> categorical <hl>. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

In particular, no first-order theory with an infinite model can be <hl> categorical <hl> .


Question: Which theorem besides the compactness and downward Lowenheim-Skolem cannot hold logic stronger than first-order

Answer: Lindström's theorem

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. <hl> This expressiveness comes at a metalogical cost, however: by Lindström's theorem , the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order. <hl>

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by <hl> Lindström's theorem <hl>, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

This expressiveness comes at a metalogical cost, however: by <hl> Lindström's theorem <hl> , the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.


Question: Which theorem talks about infinite models of first-order theory?

Answer: Löwenheim–Skolem theorem

<hl> The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. <hl> In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

The <hl> Löwenheim–Skolem theorem <hl> shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.


Question: Which theory involving infinite model cannot be categorical?

Answer: first-order

<hl> The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. <hl> In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

The Löwenheim–Skolem theorem shows that if a <hl> first-order <hl> theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.


Question: What axiomatizations of real numbers are permitted by higher-order logic?

Answer: categorical

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. <hl> In particular, no first-order theory with an infinite model can be categorical . <hl> Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be <hl> categorical <hl>. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

In particular, no first-order theory with an infinite model can be <hl> categorical <hl> .


Question: What kind of cost is incurred in higher-order logic expression?

Answer: metalogical

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. <hl> This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order. <hl>

The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a <hl> metalogical <hl> cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.


Question: Which order logic is the strongest one that can be held by compactness theorem?

Answer: first-order

<hl> The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. <hl> In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

The Löwenheim–Skolem theorem shows that if a <hl> first-order <hl> theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.


Question: What is useful as a technique to reduce the number or inference rules or axiom schemas in deductive systems?

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

Answer: Restrictions

<hl> Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. <hl> The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

<hl> Restrictions <hl> such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.


Question: There is a what between the ease of working within a formal system and the ease of proving results?

Answer: a trade-off

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. <hl> There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system. <hl>

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus <hl> a trade-off <hl> between the ease of working within the formal system and the ease of proving results about the formal system.


Question: What may be longer than derivations in systems that include additional connectives?

Answer: derivations in the limited systems

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. <hl> Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. <hl> There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, <hl> derivations in the limited systems <hl> may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.


Question: What do restrictions in deductive systems lead to?

Answer: shorter proofs of metalogical results

<hl> Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results . <hl> The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to <hl> shorter proofs of metalogical results <hl>. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to <hl> shorter proofs of metalogical results <hl> .


Question: In what systems do restrictions reduce axiom schemas?

Answer: deductive

<hl> Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. <hl> The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in <hl> deductive <hl> systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.


Question: What length of proofs incur in metalogical results by restrictions?

Answer: shorter

<hl> Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. <hl> The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to <hl> shorter <hl> proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.


Question: As and aftermath of what results in difficulty expressing natural-language statements?

Answer: restrictions

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. <hl> The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. <hl> Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the <hl> restrictions <hl> is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.


Question: What system benefits from restrictions by making results easier to prove?

Answer: deductive

<hl> Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. <hl> The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in <hl> deductive <hl> systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.


Question: Which systems derivations can be longer than additional connectives derivations?

Answer: limited

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. <hl> Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. <hl> There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the <hl> limited <hl> systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.


Question: what systems are many for first-order logic?

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

Answer: deductive systems

<hl> There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). <hl> Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many <hl> deductive systems <hl> for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.


Question: What is it called when in all models, all provable statements are true?

Answer: sound

<hl> There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). <hl> Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many deductive systems for first-order logic which are both <hl> sound <hl> (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.


Question: If in all models all statements are true and provable, what is it called?

Answer: complete

<hl> There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). <hl> Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and <hl> complete <hl> (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.


Question: The characteristic for logical consequence relation is that it is only what?

Answer: semidecidable

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). <hl> Although the logical consequence relation is only semidecidable , much progress has been made in automated theorem proving in first-order logic. <hl> First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only <hl> semidecidable <hl>, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

Although the logical consequence relation is only <hl> semidecidable <hl> , much progress has been made in automated theorem proving in first-order logic.


Question: Proving the first-order logic has made much progress made in what theorem?

Answer: automated theorem

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). <hl> Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. <hl> First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in <hl> automated theorem <hl> proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.


Question: How would "sound" be defined in first-order logic?

Answer: (all provable statements are true in all models

<hl> There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models ) and complete (all statements which are true in all models are provable). <hl> Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many deductive systems for first-order logic which are both sound <hl> (all provable statements are true in all models <hl>) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many deductive systems for first-order logic which are both sound <hl> (all provable statements are true in all models <hl> ) and complete (all statements which are true in all models are provable).


Question: How would "complete" be defined in first-order logic?

Answer: (all statements which are true in all models are provable).

<hl> There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. <hl> First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete <hl> (all statements which are true in all models are provable). <hl> Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.


Question: How would one describe logical consequence relation?

Answer: semidecidable

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). <hl> Although the logical consequence relation is only semidecidable , much progress has been made in automated theorem proving in first-order logic. <hl> First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only <hl> semidecidable <hl>, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

Although the logical consequence relation is only <hl> semidecidable <hl> , much progress has been made in automated theorem proving in first-order logic.


Question: What is one of the theorems that First-order logic satisfy?

Answer: Löwenheim–Skolem theorem

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. <hl> First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. <hl>

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the <hl> Löwenheim–Skolem theorem <hl> and the compactness theorem.


Question: In what field has progress been made via first-order logic?

Answer: much progress has been made in automated theorem proving

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). <hl> Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. <hl> First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

There are many deductive systems for first-order logic which are both sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, <hl> much progress has been made in automated theorem proving <hl> in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.


Question: What theory with an infinite domain has the strength to uniquely describe a structure?

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics.
Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic.
No first-order theory, however, has the strength to uniquely describe  a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

Answer: No first-order theory

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. <hl> No first-order theory , however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. <hl> Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. <hl> No first-order theory <hl>, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

<hl> No first-order theory <hl> , however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line.


Question: First order logic is the standard for the formalization and foundations of what?

Answer: mathematics

<hl> First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. <hl> Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

First-order logic is the standard for the formalization of <hl> mathematics <hl> into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.


Question: Just like Peano arithmetic, what other theory are the axiomatizations of number and set theory?

Answer: Zermelo–Fraenkel set theory

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. <hl> Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. <hl> No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and <hl> Zermelo–Fraenkel set theory <hl> are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.


Question: Categorical axiom systems are possible to be acquired using stronger logics such as what?

Answer: second-order logic

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. <hl> Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic . <hl>

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as <hl> second-order logic <hl>.

Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as <hl> second-order logic <hl> .


Question: What is First-order logic the standard for?

Answer: formalization of mathematics into axioms and is studied in the foundations of mathematics

<hl> First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics . <hl> Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

First-order logic is the standard for the <hl> formalization of mathematics into axioms and is studied in the foundations of mathematics <hl>. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

First-order logic is the standard for the <hl> formalization of mathematics into axioms and is studied in the foundations of mathematics <hl> .


Question: What theory is the axiomatization of number theory?

Answer: Peano arithmetic

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. <hl> Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. <hl> No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. <hl> Peano arithmetic <hl> and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.


Question: What theory is the aciomatization of set theory?

Answer: Zermelo–Fraenkel

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. <hl> Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. <hl> No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and <hl> Zermelo–Fraenkel <hl> set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.


Question: What system has the strength to uniquely describe infinite domain structures?

Answer: categorical axiom systems)

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. <hl> Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic. <hl>

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, <hl> categorical axiom systems) <hl> can be obtained in stronger logics such as second-order logic.


Question: How can one obtain complex axiom systems?

Answer: second-order logic.

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. <hl> Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic. <hl>

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as <hl> second-order logic. <hl>


Question: In the domain of discourse, an input or an output that says true or false taken by an entity or entities is called what?

A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]

Answer: predicate

<hl> A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. <hl> Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]

A <hl> predicate <hl> takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]


Question: If two sentences are viewed as unrelated under propositional logic, what can happen to them?

Answer: denoted

A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". <hl> In propositional logic, these sentences are viewed as being unrelated and might be denoted , for example, by variables such as p and q. <hl> The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]

A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated and might be <hl> denoted <hl>, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]

In propositional logic, these sentences are viewed as being unrelated and might be <hl> denoted <hl> , for example, by variables such as p and q.


Question: What does a predicate do?

Answer: takes an entity or entities in the domain of discourse as input and outputs either True or False.

<hl> A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". <hl> In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]

A predicate <hl> takes an entity or entities in the domain of discourse as input and outputs either True or False. <hl> Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]


Question: How can two sentences be denoted in propositional logic?

Answer: by variables such as p and q

A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". <hl> In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q . <hl> The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]

A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, <hl> by variables such as p and q <hl>. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]

In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, <hl> by variables such as p and q <hl> .


Question: What does the use of predicates do in regards to first-order logic?

Answer: distinguishes first-order logic from propositional logic.

A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. <hl> The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic. [clarification needed] <hl>

A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, <hl> distinguishes first-order logic from propositional logic. <hl>[clarification needed]

The use of predicates, such as "is a philosopher" in this example, <hl> distinguishes first-order logic from propositional logic. <hl> [clarification needed]


Question: How are inputs and outputs defined?

Answer: True or False.

<hl> A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". <hl> In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]

A predicate takes an entity or entities in the domain of discourse as input and outputs either <hl> True or False. <hl> Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]


Question: How can one view "Socrates is a Philosopher" vs "Plato is a philosopher" in propositional logic?

Answer: these sentences are viewed as being unrelated

A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". <hl> In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. <hl> The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]

A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, <hl> these sentences are viewed as being unrelated <hl> and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. The use of predicates, such as "is a philosopher" in this example, distinguishes first-order logic from propositional logic.[clarification needed]


Question: What is another word for "Interpretation"?

An interpretation (or model) of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.

Answer: (or model)

<hl> An interpretation (or model) of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. <hl> These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.

An interpretation <hl> (or model) <hl> of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.


Question: What does an interpretation of a first-order model specify?

Answer: what each predicate means

<hl> An interpretation (or model) of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. <hl> These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.

An interpretation (or model) of a first-order formula specifies <hl> what each predicate means <hl> and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.


Question: What do entities form the domain of?

Answer: of discourse or universe

An interpretation (or model) of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. <hl> These entities form the domain of discourse or universe , which is usually required to be a nonempty set. <hl> For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.

An interpretation (or model) of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. These entities form the domain <hl> of discourse or universe <hl>, which is usually required to be a nonempty set. For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.

These entities form the domain <hl> of discourse or universe <hl> , which is usually required to be a nonempty set.


Question: What is the domain of discourse typically required to be?

Answer: usually required to be a nonempty set.

An interpretation (or model) of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. <hl> These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato. <hl>

An interpretation (or model) of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is <hl> usually required to be a nonempty set. <hl> For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.


Question: What is another thing a first order formula can specify?

Answer: entities that can instantiate the variables

<hl> An interpretation (or model) of a first-order formula specifies what each predicate means and the entities that can instantiate the variables . <hl> These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.

An interpretation (or model) of a first-order formula specifies what each predicate means and the <hl> entities that can instantiate the variables <hl>. These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.

An interpretation (or model) of a first-order formula specifies what each predicate means and the <hl> entities that can instantiate the variables <hl> .


Question: In sufficiently expressive theories, what symbols other than function can be restricted?

It is also possible to restrict the arities of function symbols and predicate symbols, in sufficiently expressive theories. One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1 in theories that include a pairing function. This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them. It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.

Answer: arities of function symbols and predicate symbols

<hl> It is also possible to restrict the arities of function symbols and predicate symbols , in sufficiently expressive theories. <hl> One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1 in theories that include a pairing function. This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them. It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.

It is also possible to restrict the <hl> arities of function symbols and predicate symbols <hl>, in sufficiently expressive theories. One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1 in theories that include a pairing function. This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them. It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.

It is also possible to restrict the <hl> arities of function symbols and predicate symbols <hl> , in sufficiently expressive theories.


Question: In pairing function theories, functions of arity greater than what number can be dispensed?

Answer: 2

It is also possible to restrict the arities of function symbols and predicate symbols, in sufficiently expressive theories. <hl> One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1 in theories that include a pairing function. <hl> This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them. It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.

It is also possible to restrict the arities of function symbols and predicate symbols, in sufficiently expressive theories. One can in principle dispense entirely with functions of arity greater than <hl> 2 <hl> and predicates of arity greater than 1 in theories that include a pairing function. This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them. It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.


Question: In pairing function theories, predicates of arity greater than what number can be dispensed?

Answer: 1

It is also possible to restrict the arities of function symbols and predicate symbols, in sufficiently expressive theories. <hl> One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1 in theories that include a pairing function. <hl> This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them. It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.

It is also possible to restrict the arities of function symbols and predicate symbols, in sufficiently expressive theories. One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than <hl> 1 <hl> in theories that include a pairing function. This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them. It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.


Question: What kind of pair does function of arity 2 return when it involves the domain's pairs of elements?

Answer: ordered pair

It is also possible to restrict the arities of function symbols and predicate symbols, in sufficiently expressive theories. One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1 in theories that include a pairing function. <hl> This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them. <hl> It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.

It is also possible to restrict the arities of function symbols and predicate symbols, in sufficiently expressive theories. One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1 in theories that include a pairing function. This is a function of arity 2 that takes pairs of elements of the domain and returns an <hl> ordered pair <hl> containing them. It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.