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163,600 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 3 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | For a more explicit example, consider the function + which carries any pair of natural numbers to their sum. We may denote this function by \(+_{((\sigma \sigma)\sigma)}\), where \(\sigma\) is the type of natural numbers. Given any number x, \([+_{((\sigma \sigma)\sigma)}x]\) is the function which, when applied to any number y, gives the value \([[+_{((\sigma \sigma)\sigma)}x]y]\), which is ordinarily abbreviated as \(x + y\). Thus \([+_{((\sigma \sigma)\sigma)}x]\) is the function of one argument which adds x to any number. When we think of \(+_{((\sigma \sigma)\sigma)}\) as a function of one argument, we see that it maps any number x to the function \([+_{((\sigma \sigma)\sigma)}x]\). |
163,601 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 4 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | More generally, if f is a function which maps n-tuples \(\langle w_{\beta},x_{\gamma},\ldots ,y_{\delta},z_{\tau}\rangle\) of elements of types \(\beta\), \(\gamma\),…, \(\delta\) ,\(\tau\), respectively, to elements of type α, we may assign to f the type \(((\ldots((\alpha \tau)\delta)\ldots \gamma)\beta)\). It is customary to use the convention of association to the left to omit parentheses, and write this type symbol simply as \((\alpha \tau \delta \ldots \gamma \beta)\). |
163,602 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 5 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | A set or property can be represented by a function (often called characteristic function) which maps elements to truth values, so that an element is in the set, or has the property, in question iff the function representing the set or property maps that element to truth. When a statement is asserted, the speaker means that it is true, so that \(s x\) means that \(s x\) is true, which also expresses the assertions that s maps x to truth and that \(x \in s\). In other words, \(x \in s\) iff \(s x\). We take \({o}\) as the type symbol denoting the type of truth values, so we may speak of any function of type \(({o}\alpha)\) as a set of elements of type α. A function of type \((({o}\alpha)\beta)\) is a binary relation between elements of type β and elements of type α. For example, if \(\sigma\) is the type of the natural numbers, and \(<\) is the order relation between natural numbers, \(<\) has type \(({o}\sigma \sigma)\), and for all natural numbers x and \(y, {<}x y\) (which we ordinarily write as \(x < y)\) has the value truth iff x is less than y. Of course, \(<\) can also be regarded as the function which maps each natural number x to the set \(<x\) of all natural numbers y such that x is less than y. Thus sets, properties, and relations may be regarded as particular kinds of functions. Church’s type type theory is thus a logic of functions, and, in this sense, it is in the tradition of the work of Frege’s Begriffsschrift. The opposite approach would be to reduce functions to relations, which was the approach taken by Whitehead and Russell (1927a) in the Principia Mathematica. |
163,603 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 6 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Expressions which denote elements of type α are called wffs of type α. Thus, statements of type theory are wffs of type \({o}\). |
163,604 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 7 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | If \(\bA_{\alpha}\) is a wff of type α in which \(\bu_{\alpha \beta}\) is not free, the function (associated with) \(\bu_{\alpha \beta}\) such that \(\forall \bv_{\beta}[\bu_{\alpha \beta}\bv_{\beta} = \bA_{\alpha}]\) is denoted by \([\lambda \bv_{\beta}\bA_{\alpha}]\). Thus \(\lambda \bv_{\beta}\) is a variable-binder, like \(\forall \bv_{\beta}\) or \(\exists \bv_{\beta}\) (but with a quite different meaning, of course); λ is known as an abstraction operator. \([\lambda \bv_{\beta}\bA_{\alpha}]\) denotes the function whose value on any argument \(\bv_{\beta}\) is \(\bA_{\alpha}\), where \(\bv_{\beta}\) may occur free in \(\bA_{\alpha}\). For example, \([\lambda n_{\sigma}[4\cdot n_{\sigma}+3]]\) denotes the function whose value on any natural number n is \(4\cdot n+3\). Hence, when we apply this function to the number 5 we obtain \([\lambda n_{\sigma}[4\cdot n_{\sigma}+3]]5 = 4\cdot 5+3 = 23\). |
163,605 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 8 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | We use \(\textsf{Sub}(\bB,\bv,\bA)\) as a notation for the result of substituting \(\bB\) for \(\bv\) in \(\bA\), and \(\textsf{SubFree}(\bB,\bv,\bA)\) as a notation for the result of substituting \(\bB\) for all free occurrences of \(\bv\) in \(\bA\). The process of replacing \([\lambda \bv_{\beta}\bA_{\alpha}]\bB_{\beta}\) by \(\textsf{SubFree}(\bB_{\beta},\bv_{\beta},\bA_{\alpha})\) (or vice-versa) is known as β-conversion, which is one form of λ-conversion. Of course, when \(\bA_{{o}}\) is a wff of type \({o}\), \([\lambda \bv_{\beta}\bA_{{o}}]\) denotes the set of all elements \(\bv_{\beta}\) (of type \(\beta)\) of which \(\bA_{{o}}\) is true; this set may also be denoted by \(\{\bv_{\beta}|\bA_{{o}}\}\). For example, \([\lambda x\ x<y]\) denotes the set of x such that x is less than y (as well as that property which a number x has if it is less than y). In familiar set-theoretic notation, |
163,606 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 9 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | would be written |
163,607 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 10 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | (By the Axiom of Extensionality for truth values, when \(\bC_{{o}}\) and \(\bD_{{o}}\) are of type \({o}, \bC_{{o}} \equiv \bD_{{o}}\) is equivalent to \(\bC_{{o}} = \bD_{{o}}\).) |
163,608 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 11 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Propositional connectives and quantifiers can be assigned types and can be denoted by constants of these types. The negation function maps truth values to truth values, so it has type \(({o}{o})\). Similarly, disjunction and conjunction (etc.) are binary functions from truth values to truth values, so they have type \(({o}{o}{o})\). |
163,609 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 12 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The statement \(\forall \bx_{\alpha}\bA_{{o}}\) is true iff the set \([\lambda \bx_{\alpha}\bA_{{o}}]\) contains all elements of type α. A constant \(\Pi_{{o}({o}\alpha)}\) can be introduced (for each type symbol \(\alpha)\) to denote a property of sets: a set \(s_{{o}\alpha}\) has the property \(\Pi_{{o}({o}\alpha)}\) iff \(s_{{o}\alpha}\) contains all elements of type α. With this interpretation |
163,610 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 13 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | should be true, as well as |
163,611 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 14 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | for any wff \(\bA_{{o}}\) and variable \(\bx_{\alpha}\). Since by λ-conversion we have |
163,612 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 15 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | equation can be written more simply as |
163,613 | Church’s Type Theory | 1. Syntax | 1.1 Fundamental Ideas | 16 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Thus, \(\forall \bx_{\alpha}\) can be defined in terms of \(\Pi_{{o}({o}\alpha)}\), and λ is the only variable-binder that is needed. |
163,614 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 0 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Before we state the definition of a “formula”, a word of caution is in order. The reader may be accustomed to thinking of a formula as an expression which plays the role of an assertion in a formal language, and of a term as an expression which designates an object. Church’s terminology is somewhat different, and provides a uniform way of discussing expressions of many different types. What we call well-formed formula of type α (\(\textrm{wff}_{\alpha}\)) below would in more standard terminology be called term of type α, and then only certain terms, namely those with type \({o}\), would be called formulas. Anyhow, in this entry we have decided to stay with Church’s original terminology. Another remark concerns the use of some specific mathematical notation. In what follows, the entry distinguishes between the symbols \(\imath\), \(\iota_{(\alpha({o}\alpha))}\), and \(\atoi\). The first is the symbol used for the type of individuals; the second is the symbol used for a logical constant (see Section 1.2.1 below); the third is the symbol used as a variable-binding operator that represents the definite description “the” (see Section 1.3.4). The reader should not confuse them and check to see that the browser is displaying these symbols correctly. |
163,615 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 1 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Type symbols are defined inductively as follows: |
163,616 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 2 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The primitive symbols are the following: |
163,617 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 3 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | A formula is a finite sequence of primitive symbols. Certain formulas are called well-formed formulas (wffs). We write \(\textrm{wff}_{\alpha}\) as an abbreviation for wff of type α, and define this concept inductively as follows: |
163,618 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 4 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Note, for example, that by (a) \(\nsim_{({o}{o})}\) is a wff\(_{({o}{o})}\), so by (b) if \(\bA_{{o}}\) is a wff\(_{{o}}\), then \([\nsim_{({o}{o})}\bA_{{o}}]\) is a wff\(_{{o}}\). Usually, the latter wff will simply be written as \(\nsim \bA\). It is often convenient to avoid parentheses, brackets and type symbols, and use conventions for omitting them. For formulas we use the convention of association to the right, and we may write \(\lor_{((oo)o)}\bA_{{o}} \bB_{{o}}\) instead of \([[\lor_{((oo)o)}\bA_{{o}}] \bB_{{o}}]\). For types the corresponding convention is association to the left, and we may write \(ooo\) instead of \(((oo)o)\). |
163,619 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 5 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The last definition is known as the Leibnizian definition of equality. It asserts that x and y are the same if y has every property that x has. Actually, Leibniz called his definition “the identity of indiscernibles” and gave it in the form of a biconditional: x and y are the same if x and y have exactly the same properties. It is not difficult to show that these two forms of the definition are logically equivalent. |
163,620 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 6 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | We now provide a few examples to illustrate how various assertions and concepts can be expressed in Church’s type theory. |
163,621 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 7 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Example 1 To express the assertion that “Napoleon is charismatic” we introduce constants \(\const{Charismatic}_{{o}\imath}\) and \(\const{Napoleon}_{\imath}\), with the types indicated by their subscripts and the obvious meanings, and assert the wff |
163,622 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 8 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | If we wish to express the assertion that “Napoleon has all the properties of a great general”, we might consider interpreting this to mean that “Napoleon has all the properties of some great general”, but it seems more appropriate to interpret this statement as meaning that “Napoleon has all the properties which all great generals have”. If the constant \(\const{GreatGeneral}_{{o}\imath}\) is added to the formal language, this can be expressed by the wff |
163,623 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 9 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | As an example of such a property, we note that the sentence “Napoleon’s soldiers admire him” can be expressed in a similar way by the wff |
163,624 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 10 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | By λ-conversion, this is equivalent to |
163,625 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 11 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | This statement asserts that one of the properties which Napoleon has is that of being admired by his soldiers. The property itself is expressed by the wff |
163,626 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 12 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Example 2 We illustrate some potential applications of type theory with the following fable. |
163,627 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 13 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | A rich and somewhat eccentric lady named Sheila has an ostrich and a cheetah as pets, and she wishes to take them from her hotel to her remote and almost inaccessible farm. Various portions of the trip may involve using elevators, boxcars, airplanes, trucks, very small boats, donkey carts, suspension bridges, etc., and she and the pets will not always be together. She knows that she must not permit the ostrich and the cheetah to be together when she is not with them. |
163,628 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 14 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | We consider how certain aspects of this problem can be formalized so that Sheila can use an automated reasoning system to help analyze the possibilities. |
163,629 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 15 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | There will be a set Moments of instants or intervals of time during the trip. She will start the trip at the location \(\const{Hotel}\) and moment \(\const{Start}\), and end it at the location \(\const{Farm}\) and moment \(\const{Finish}\). Moments will have type \(\tau\), and locations will have type \(\varrho\). A state will have type \(\sigma\) and will specify the location of Sheila, the ostrich, and the cheetah at a given moment. A plan will specify where the entities will be at each moment according to this plan. It will be a function from moments to states, and will have type \((\sigma \tau)\). The exact representation of states need not concern us, but there will be functions from states to locations called \(\const{LocationOfSheila}\), \(\const{LocationOfOstrich}\), and \(\const{LocationOfCheetah}\) which provide the indicated information. Thus, \(\const{LocationOfSheila}_{\varrho \sigma}[p_{\sigma \tau}t_{\tau}]\) will be the location of Sheila according to plan \(p_{\sigma \tau}\) at moment \(t_{\tau}\). The set \(\const{Proposals}_{{o}(\sigma \tau)}\) is the set of plans Sheila is considering. |
163,630 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 16 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | We define a plan p to be acceptable if, according to that plan, the group starts at the hotel, finishes at the farm, and whenever the ostrich and the cheetah are together, Sheila is there too. Formally, we define \(\const{Acceptable}_{{o}(\sigma \tau)}\) as |
163,631 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 17 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | We can express the assertion that Sheila has a way to accomplish her objective with the formula |
163,632 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 18 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Example 3 We now provide a mathematical example. Mathematical ideas can be expressed in type theory without introducing any new constants. An iterate of a function f from a set to itself is a function which applies f one or more times. For example, if \(g(x) = f(f(f(x)))\), then g is an iterate of f. \([\text{ITERATE+}_{{o}(\imath\imath)(\imath\imath)}f_{\imath\imath}g_{\imath\imath}]\) means that \(g_{\imath\imath}\) is an iterate of \(f_{\imath\imath}\). \(\text{ITERATE+}_{{o}(\imath\imath)(\imath\imath)}\) is defined (inductively) as |
163,633 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 19 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Thus, g is an iterate of f if g is in every set p of functions which contains f and which contains the function \(\lambda x_{\imath}f_{\imath\imath}[j_{\imath\imath}x_{\imath}]\) (i.e., f composed with j) whenever it contains j. |
163,634 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 20 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | A fixed point of f is an element y such that \(f(y) = y\). |
163,635 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 21 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | It can be proved that if some iterate of a function f has a unique fixed point, then f itself has a fixed point. This theorem can be expressed by the wff |
163,636 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 22 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | See Andrews et al. 1996, for a discussion of how this theorem, which is called THM15B, can be proved automatically. |
163,637 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 23 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Example 4 An example from philosophy is Gödel’s variant of the ontological argument for the existence of God. This example illustrates two interesting aspects: |
163,638 | Church’s Type Theory | 1. Syntax | 1.2 Formulas | 24 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Example 5 Suppose we omit the use of type symbols in the definitions of wffs. Then we can write the formula \(\lambda x\nsim[xx]\), which we shall call \(\textrm{R}\). It can be regarded as denoting the set of all sets x such that x is not in x. We may then consider the formula \([\textrm{R R}]\), which expresses the assertion that \(\textrm{R}\) is in itself. We can clearly prove \([\textrm{R R}] \equiv [[\lambda x\nsim [xx]] \textrm{R}]\), so by λ-conversion we can derive \([\textrm{R R}] \equiv\, \nsim[\textrm{R R}]\), which is a contradiction. This is Russell’s paradox. Russell’s discovery of this paradox (Russell 1903, 101-107) played a crucial role in the development of type theory. Of course, when type symbols are present, \(\textrm{R}\) is not well-formed, and the contradiction cannot be derived. |
163,639 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 0 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | We start by listing the axioms for what we shall call elementary type theory. |
163,640 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 1 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The theorems of elementary type theory are those theorems which can be derived, using the rules of inference, from Axioms (1)–\((6^{\alpha})\) (for all type symbols \(\alpha)\). We shall sometimes refer to elementary type theory as \(\cT\). It embodies the logic of propositional connectives, quantifiers, and λ-conversion in the context of type theory. |
163,641 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 2 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | To illustrate the rules and axioms introduced above, we give a short and trivial proof in \(\cT\). Following each wff of the proof, we indicate how it was inferred. (The proof is actually quite inefficient, since line 3 is not used later, and line 7 can be derived directly from line 5 without using line 6. The additional proof lines have been inserted to illustrate some relevant aspects. For the sake of readability, many brackets have been deleted from the formulas in this proof. The diligent reader should be able to restore them.) |
163,642 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 3 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Note that (3) can be written as |
163,643 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 4 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | and (7) can be written as |
163,644 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 5 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | We have thus derived a well known law of quantification theory. We illustrate one possible interpretation of the wff \((7')\) (which is closely related to Axiom 6) by considering a situation in which a rancher puts some horses in a corral and leaves for the night. Later, he cannot remember whether he closed the gate to the corral. While reflecting on the situation, he comes to a conclusion which can be expressed by \((7')\) if we take the horses to be the elements of type \(\imath\), interpret \(p_{{o}}\) to mean “the gate was closed”, and interpret \(r_{{o}\imath}\) so that \(r_{{o}\imath}x_{\imath}\) asserts “\(x_{\imath}\) left the corral”. With this interpretation, \((7')\) says |
163,645 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 6 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | If it is true of every horse that the gate was closed or that the horse left the corral, then the gate was closed or every horse left the corral. |
163,646 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 7 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | To the axioms listed above we add the axioms below to obtain Church’s type theory. |
163,647 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 8 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The axioms of boolean and functional extensionality are the following: |
163,648 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 9 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Church did not include Axiom \(7^{{o}}\) in his list of axioms in Church 1940, but he mentioned the possibility of including it. Henkin did include it in Henkin 1950. |
163,649 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 10 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The expression |
163,650 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 11 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | stands for |
163,651 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 12 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | For example, |
163,652 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 13 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | stands for |
163,653 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 14 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | By λ-conversion, this is equivalent to |
163,654 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 15 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | which reduces by λ-conversion to |
163,655 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 16 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | This asserts that there is a unique element which has the property \(P_{{o}\alpha}\). From this example we can see that in general, \(\exists_1\bx_{\alpha}\bA_{{o}}\) expresses the assertion that “there is a unique \(\bx_{\alpha}\) such that \(\bA_{{o}}\)”. |
163,656 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 17 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | When there is a unique such element \(\bx_{\alpha}\), it is convenient to have the notation \(\atoi\bx_{\alpha}\bA_{{o}}\) to represent the expression “the \(\bx_{\alpha}\) such that \(\bA_{{o}}\)”. Russell showed in Whitehead & Russell 1927b how to provide contextual definitions for such notations in his formulation of type theory. In Church’s type theory \(\atoi\bx_{\alpha}\bA_{{o}}\) is defined as \(\iota_{\alpha({o}\alpha)}[\lambda \bx_{\alpha}\bA_{{o}}]\). Thus, \(\atoi\) behaves like a variable-binding operator, but it is defined in terms of λ with the aid of the constant \(\iota_{\alpha({o}\alpha)}\). Thus, λ is still the only variable-binding operator that is needed. |
163,657 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 18 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Since \(\bA_{{o}}\) describes \(\bx_{\alpha}, \iota_{\alpha({o}\alpha)}\) is called a description operator. Associated with this notation is the following: |
163,658 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 19 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | This says that when the set \(p_{{o}\alpha}\) has a unique member, then \(\iota_{\alpha({o}\alpha)}p_{{o}\alpha}\) is in \(p_{{o}\alpha}\), and therefore is that unique member. Thus, this axiom asserts that \(\iota_{\alpha({o}\alpha)}\) maps one-element sets to their unique members. |
163,659 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 20 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | If from certain hypotheses one can prove |
163,660 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 21 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | then by using Axiom \(8^{\alpha}\) one can derive |
163,661 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 22 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | which can also be written as |
163,662 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 23 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | We illustrate the usefulness of the description operator with a small example. Suppose we have formalized the theory of real numbers, and our theory has constants \(1_{\varrho}\) and \(\times_{\varrho \varrho \varrho}\) to represent the number 1 and the multiplication function, respectively. (Here \(\varrho\) is the type of real numbers.) To represent the multiplicative inverse function, we can define the wff \(\textrm{INV}_{\varrho \varrho}\) as |
163,663 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 24 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Of course, in traditional mathematical notation we would not write the type symbols, and we would write \(\times_{\varrho \varrho \varrho}z_{\varrho}x_{\varrho}\) as \(z \times x\) and write \(\textrm{INV}_{\varrho \varrho}z\) as \(z^{-1}\). Thus \(z^{-1}\) is defined to be that x such that \(z \times x = 1\). When Z is provably not 0, we will be able to prove \(\exists_1 x_{\varrho}[\times_{\varrho \varrho \varrho} \textrm{Z x}_{\varrho} = 1_{\varrho}]\) and \(Z \times Z^{-1} = 1\), but if we cannot establish that Z is not 0, nothing significant about \(Z^{-1}\) will be provable. |
163,664 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 25 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The Axiom of Choice can be expressed as follows in Church’s type theory: |
163,665 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 26 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | \((9^{\alpha})\) says that the choice function \(\iota_{\alpha({o}\alpha)}\) chooses from every nonempty set \(p_{{o}\alpha}\) an element, designated as \(\iota_{\alpha({o}\alpha)}p_{{o}\alpha}\), of that set. When this form of the Axiom of Choice is included in the list of axioms, \(\iota_{\alpha({o}\alpha)}\) is called a selection operator instead of a description operator, and \(\atoi\bx_{\alpha} \bA_{{o}}\) means “an \(\bx_{\alpha}\) such that \(\bA_{{o}}\)” when there is some such element \(\bx_{\alpha}\). These selection operators have the same meaning as Hilbert’s \(\epsilon\)-operator (Hilbert 1928). However, we here provide one such operator for each type α. |
163,666 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 27 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | It is natural to call \(\atoi\) a definite description operator in contexts where \(\atoi\bx_{\alpha}\bA_{{o}}\) means “the \(\bx_{\alpha}\) such that \(\bA_{{o}}\)”, and to call it an indefinite description operator in contexts where \(\atoi\bx_{\alpha}\bA_{{o}}\) means “an \(\bx_{\alpha}\) such that \(\bA_{{o}}\)”. |
163,667 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 28 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Clearly the Axiom of Choice implies the Axiom of Descriptions, but sometimes formulations of type theory are used which include the Axiom of Descriptions, but not the Axiom of Choice. |
163,668 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 29 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Another formulation of the Axiom of Choice simply asserts the existence of a choice function without explicitly naming it: |
163,669 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 30 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Normally when one assumes the Axiom of Choice in type theory, one assumes it as an axiom schema, and asserts AC\(^{\alpha}\) for each type symbol α. A similar remark applies to the axioms for extensionality and description. However, modern proof systems for Church’s type theory, which are, e.g., based on resolution, do in fact avoid the addition of such axiom schemata for reasons as further explained in Sections 3.4 and 4 below. They work with more constrained, goal-directed proof rules instead. |
163,670 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 31 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Before proceeding, we need to introduce some terminology. \(\cQ_0\) is an alternative formulation of Church’s type theory which will be described in Section 1.4 and is equivalent to the system described above using Axioms (1)–(8). A type symbol is propositional if the only symbols which occur in it are \({o}\) and parentheses. |
163,671 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 32 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Yasuhara (1975) defined the relation “\(\ge\)” between types as the reflexive transitive closure of the minimal relation such that \((\alpha \beta) \ge \alpha\) and \((\alpha \beta) \ge \beta\). He established that: |
163,672 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 33 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The existence of a choice functions for “higher” types thus entails the existence of choice functions for “lower” types, the opposite is generally not the case though. |
163,673 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 34 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Büchi (1953) has shown that while the schemas expressing the Axiom of Choice and Zorn’s Lemma can be derived from each other, the relationships between the particular types involved are complex. |
163,674 | Church’s Type Theory | 1. Syntax | 1.3 Axioms and Rules of Inference | 35 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | One can define the natural numbers (and therefore other basic mathematical structures such as the real and complex numbers) in type theory, but to prove that they have the required properties (such as Peano’s Postulates), one needs an Axiom of Infinity. There are many viable possibilities for such an axiom, such as those discussed in Church 1940, section 57 of Church 1956, and section 60 of Andrews 2002. |
163,675 | Church’s Type Theory | 1. Syntax | 1.4 A Formulation Based on Equality | 0 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | In Section 1.2.1, \(\nsim_{({o}{o})}, \lor_{(({o}{o}){o})}\), and the \(\Pi_{({o}({o}\alpha))}\)’s were taken as primitive constants, and the wffs \(\sfQ_{{o}\alpha \alpha}\) which denote equality relations at type α were defined in terms of these. We now present an alternative formulation \(\cQ_0\) of Church’s type theory in which there are primitive constants \(\sfQ_{{o}\alpha \alpha}\) denoting equality, and \(\nsim_{({o}{o})}, \lor_{(({o}{o}){o})}\), and the \(\Pi_{({o}({o}\alpha))}\)’s are defined in terms of the \(\sfQ_{{o}\alpha \alpha}\)’s. |
163,676 | Church’s Type Theory | 1. Syntax | 1.4 A Formulation Based on Equality | 1 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Tarski (1923) noted that in the context of higher-order logic, one can define propositional connectives in terms of logical equivalence and quantifiers. Quine (1956) showed how both quantifiers and connectives can be defined in terms of equality and the abstraction operator λ in the context of Church’s type theory. Henkin (1963) rediscovered these definitions, and developed a formulation of Church’s type theory based on equality in which he restricted attention to propositional types. Andrews (1963) simplified the axioms for this system. |
163,677 | Church’s Type Theory | 1. Syntax | 1.4 A Formulation Based on Equality | 2 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | \(\cQ_0\) is based on these ideas, and can be shown to be equivalent to a formulation of Church’s type theory using Axioms (1)–(8) of the preceding sections. This section thus provides an alternative to the material in the preceding Sections 1.2.1–1.3.4. More details about \(\cQ_0\) can be found in Andrews 2002. |
163,678 | Church’s Type Theory | 1. Syntax | 1.4 A Formulation Based on Equality | 3 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | \(T_{{o}}\) denotes truth. The meaning of \(\Pi_{{o}({o}\alpha)}\) was discussed in Section 1.1. To see that this definition of \(\Pi_{{o}({o}\alpha)}\) is appropriate, note that \(\lambda x_{\alpha}T\) denotes the set of all elements of type α, and that \(\Pi_{{o}({o}\alpha)}s_{{o}\alpha}\) stands for \(\sfQ_{{o}({o}\alpha)({o}\alpha)}[\lambda x_{\alpha}T] s_{{o}\alpha}\), respectively for \([\lambda x_{\alpha}T] = s_{{o}\alpha}\). Therefore \(\Pi_{{o}({o}\alpha)}s_{{o}\alpha}\) asserts that \(s_{{o}\alpha}\) is the set of all elements of type α, so \(s_{{o}\alpha}\) contains all elements of type α. It can be seen that \(F_{{o}}\) can also be written as \(\forall x_{{o}}x_{{o}}\), which asserts that everything is true. This is false, so \(F_{{o}}\) denotes falsehood. The expression \(\lambda g_{{o}{o}{o}}[g_{{o}{o}{o}}x_{{o}}y_{{o}}]\) can be used to represent the ordered pair \(\langle x_{{o}},y_{{o}}\rangle\), and the conjunction \(x_{{o}} \land y_{{o}}\) is true iff \(x_{{o}}\) and \(y_{{o}}\) are both true, i.e., iff \(\langle T_{{o}},T_{{o}}\rangle = \langle x_{{o}},y_{{o}}\rangle\). Hence \(x_{{o}} \land y_{{o}}\) can be expressed by the formula \([\lambda g_{{o}{o}{o}}[g_{{o}{o}{o}}T_{{o}}T_{{o}}]] = [\lambda g_{{o}{o}{o}}[g_{{o}{o}{o}}x_{{o}}y_{{o}}]]\). |
163,679 | Church’s Type Theory | 1. Syntax | 1.4 A Formulation Based on Equality | 4 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Other propositional connectives and the existential quantifier are easily defined. By using \(\iota_{(\imath({o}\imath))}\), one can define description operators \(\iota_{\alpha({o}\alpha)}\) for all types α. |
163,680 | Church’s Type Theory | 1. Syntax | 1.4 A Formulation Based on Equality | 5 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | \(\cQ_0\) has a single rule of inference. |
163,681 | Church’s Type Theory | 1. Syntax | 1.4 A Formulation Based on Equality | 6 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Rule R: From \(\bC\) and \(\bA_{\alpha} = \bB_{\alpha}\), to infer the result of replacing one occurrence of \(\bA_{\alpha}\) in \(\bC\) by an occurrence of \(\bB_{\alpha}\), provided that the occurrence of \(\bA_{\alpha}\) in \(\bC\) is not (an occurrence of a variable) immediately preceded by λ. |
163,682 | Church’s Type Theory | 1. Syntax | 1.4 A Formulation Based on Equality | 7 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The axioms for \(\cQ_0\) are the following: |
163,683 | Church’s Type Theory | 2. Semantics | null | 0 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | It is natural to compare the semantics of type theory with the semantics of first-order logic, where the theorems are precisely the wffs which are valid in all interpretations. From an intuitive point of view, the natural interpretations of type theory are standard models, which are defined below. However, it is a consequence of Gödel’s Incompleteness Theorem (Gödel 1931) that axioms (1)–(9) do not suffice to derive all wffs which are valid in all standard models, and there is no consistent recursively axiomatized extension of these axioms which suffices for this purpose. Nevertheless, experience shows that these axioms are sufficient for most purposes, and Leon Henkin considered the problem of clarifying in what sense they are complete. The definitions and theorem below constitute Henkin’s (1950) solution to this problem, which is often referred to as general semantics or Henkin semantics. |
163,684 | Church’s Type Theory | 2. Semantics | null | 1 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | A frame is a collection \(\{\cD_{\alpha}\}_{\alpha}\) of nonempty domains (sets) \(\cD_{\alpha}\), one for each type symbol α, such that \(\cD_{{o}} = \{\sfT,\sfF\}\) (where \(\sfT\) represents truth and \(\sfF\) represents falsehood), and \(\cD_{\alpha \beta}\) is some collection of functions mapping \(\cD_{\beta}\) into \(\cD_{\alpha}\). The members of \(\cD_{\imath}\) are called individuals. |
163,685 | Church’s Type Theory | 2. Semantics | null | 2 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | An interpretation \(\langle \{\cD_{\alpha}\}_{\alpha}, \frI\rangle\) consists of a frame and a function \(\frI\) which maps each constant C of type α to an appropriate element of \(\cD_{\alpha}\), which is called the denotation of C. The logical constants are given their standard denotations. |
163,686 | Church’s Type Theory | 2. Semantics | null | 3 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | An assignment of values in the frame \(\{\cD_{\alpha}\}_{\alpha}\) to variables is a function \(\phi\) such that \(\phi \bx_{\alpha} \in \cD_{\alpha}\) for each variable \(\bx_{\alpha}\). (Notation: The assignment \(\phi[a/x]\) maps variable x to value a and it is identical with \(\phi\) for all other variable symbols different from x.) |
163,687 | Church’s Type Theory | 2. Semantics | null | 4 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | An interpretation \(\cM = \langle \{\cD_{\alpha}\}_{\alpha}, \frI\rangle\) is a general model (aka Henkin model) iff there is a binary function \(\cV\) such that \(\cV_{\phi}\bA_{\alpha} \in \cD_{\alpha}\) for each assignment \(\phi\) and wff \(\bA_{\alpha}\), and the following conditions are satisfied for all assignments and all wffs: |
163,688 | Church’s Type Theory | 2. Semantics | null | 5 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | If an interpretation \(\cM\) is a general model, the function \(\cV\) is uniquely determined. \(\cV_{\phi}\bA_{\alpha}\) is called the value of \(\bA_{\alpha}\) in \(\cM\) with respect to \(\phi\). |
163,689 | Church’s Type Theory | 2. Semantics | null | 6 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | One can easily show that the following statements hold in all general models \(\cM\) for all assignments \(\phi\) and all wffs \(\bA\) and \(\bB\): |
163,690 | Church’s Type Theory | 2. Semantics | null | 7 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | The semantics of general models is thus as expected. However, there is a subtlety to note regarding the following condition for arbitrary types α: |
163,691 | Church’s Type Theory | 2. Semantics | null | 8 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | When the definitions of Section 1.2.1 are employed, where equality has been defined in terms of Leibniz’ principle, then this statement is not implied for all types α. It only holds if we additionally require that the domains \(\cD_{{o}\alpha}\) contain all the unit sets of objects of type α, or, alternatively, that the domains \(\cD_{{o}\alpha\alpha}\) contain the respective identity relations on objects of type α (which entails the former). The need for this additional requirement, which is not included in the original work of Henkin (1950), has been demonstrated in Andrews 1972a. |
163,692 | Church’s Type Theory | 2. Semantics | null | 9 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | When instead the alternative definitions of Section 1.4 are employed, then this requirement is obviously met due to the presence of the logical constants \(\sfQ_{{o}\alpha \alpha}\) in the signature, which by definition denote the respective identity relations on the objects of type α and therefore trivially ensure their existence in each general model \(\cM\). It is therefore a natural option to always assume primitive equality constants (for each type α) in a concrete choice of base system for Church’s type theory, just as realized in Andrews’ system \(\cQ_0\). |
163,693 | Church’s Type Theory | 2. Semantics | null | 10 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | An interpretation \(\langle \{\cD_{\alpha}\}_{\alpha}, \frI\rangle\) is a standard model iff for all α and \(\beta , \cD_{\alpha \beta}\) is the set of all functions from \(\cD_{\beta}\) into \(\cD_{\alpha}\). Clearly a standard model is a general model. |
163,694 | Church’s Type Theory | 2. Semantics | null | 11 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | We say that a wff \(\bA\) is valid in a model \(\cM\) iff \(\cV_{\phi}\bA = \sfT\) for every assignment \(\phi\) into \(\cM\). A model for a set \(\cH\) of wffs is a model in which each wff of \(\cH\) is valid. |
163,695 | Church’s Type Theory | 2. Semantics | null | 12 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | A wff \(\bA\) is valid in the general [standard] sense iff \(\bA\) is valid in every general [standard] model. Clearly a wff which is valid in the general sense is valid in the standard sense, but the converse of this statement is false. |
163,696 | Church’s Type Theory | 2. Semantics | null | 13 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Completeness and Soundness Theorem (Henkin 1950): A wff is a theorem if and only if it is valid in the general sense. |
163,697 | Church’s Type Theory | 2. Semantics | null | 14 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | Not all frames belong to interpretations, and not all interpretations are general models. In order to be a general model, an interpretation must have a frame satisfying certain closure conditions which are discussed further in Andrews 1972b. Basically, in a general model every wff must have a value with respect to each assignment. |
163,698 | Church’s Type Theory | 2. Semantics | null | 15 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | A model is said to be finite iff its domain of individuals is finite. Every finite model for \(\cQ_0\) is standard (Andrews 2002, Theorem 5404), but every set of sentences of \(\cQ_0\) which has infinite models also has nonstandard models (Andrews2002, Theorem 5506). |
163,699 | Church’s Type Theory | 2. Semantics | null | 16 | type-theory-church | First published Fri Aug 25, 2006; substantive revision Tue May 21, 2019 | An understanding of the distinction between standard and nonstandard models can clarify many phenomena. For example, it can be shown that there is a model \(\cM = \langle \{\cD_{\alpha}\}_{\alpha}, \frI\rangle\) in which \(\cD_{\imath}\) is infinite, and all the domains \(\cD_{\alpha}\) are countable. Thus \(\cD_{\imath}\) and \(\cD_{{o}\imath}\) are both countably infinite, so there must be a bijection h between them. However, Cantor’s Theorem (which is provable in type theory and therefore valid in all models) says that \(\cD_{\imath}\) has more subsets than members. This seemingly paradoxical situation is called Skolem’s Paradox. It can be resolved by looking carefully at Cantor’s Theorem, i.e., \(\nsim \exists g_{{o}\imath\imath}\forall f_{{o}\imath}\exists j_{\imath}[g_{{o}\imath\imath}j_{\imath} = f_{{o}\imath}]\), and considering what it means in a model. The theorem says that there is no function \(g \in \cD_{{o}\imath\imath}\) from \(\cD_{\imath}\) into \(\cD_{{o}\imath}\) which has every set \(f_{{o}\imath} \in \cD_{{o}\imath}\) in its range. The usual interpretation of the statement is that \(\cD_{{o}\imath}\) is bigger (in cardinality) than \(\cD_{\imath}\). However, what it actually means in this model is that h cannot be in \(\cD_{{o}\imath\imath}\). Of course, \(\cM\) must be nonstandard. |