Talk:Gödel's incompleteness theorems/Archive 1

message: From Eginhart Biedermann,   biedermann@clix.pt       15.9.2005
Biedermann 12:39, 13 October 2005 (UTC)
thread_title: 1

message: [COMMENT: So how is this relevant to Objectivism? Does Objectivism have
the aim of proving all mathematical truths?]
:I don't even know what Objectivism is :-) --AxelBoldt
:: Objectivism is a philosophical system created by Ayn Rand. It is heavily based on the work of Aristotle. The two viewpoints are irreconcilable in terms of mathematics (see axiom 1 of the Objectivist page), but then Objectivism is as much a "framework for living" as anything else, and Goedel didn't directly address "self-esteem" in any of his work. - MB
::: Objectivism does not pretend that reason itself is formalizable. Goedel's theorem applies only to formal systems. This is the difference between reason and logic. There are many forms of logic, all equally true in the sense of being consistent, and with varying arenas of applicability or inapplicability. Symbols in logic do not 'mean' anything in reality, they just refer to other logical placeholders. Objectivism, however, is not a logic or formal system, it's manipulation of meaning where there are no placeholders. I have a larger article on this somewhere which I can find and link if there's interest, but suffice it to say that Goedel, if anything, helps to show why Objectivism is right that no logical system can substitute for reason. In any event, at minimum, Objectivists have convinced themselves that Goedel is no problem for them. - JoelKatz
:I hear this about this "problem" all the time but I have yet to hear a single decent argument on how the theorem affects Objectivism, much less refutes anything...  --WayneMokane 07:16, 19 Dec 2004 (UTC)
thread_title: Objectivism

message: Dan Willard showed a few years back that there are a family of consistent systems over the language of first-order arithmetic that are capable of proving their own consistency.  I'm afraid there are no good online intrductions to the idea, but I can trace the outline:
* The key is to formalise enough of the Goedel machinery to talk about provability internally without being able to formalise diagonalisation;
* Diagonalisation depends upon not being able to prove multiplication total (and in the earlier versions of the result, addition also);
* Addition and multiplication are not function symbols of the language, instead subtraction and division are, with the addition and multiplication predicates being defined in terms of these.  The we can't prove (All x,y)(Exists z) multiply(x,y,z).
* Totality of muliplication is a Pi-0-2 sentence;
* Provability can be expressed in terms of a tableau search algorithm;
* Provability of consistency can then simply be added as an axiom.  The resulting system can be proven consistent by means of a relative consistency argument wrt. regular arithmetic.
* We can add any valid Pi-0-1 sentence of arithmetic to the theory and still remain consistent.
Given the above result, we have to abandon the claim:
:Gödel's second incompleteness theorem, which is proved by formalizing part of the proof of the first within the system itself, states:
::No consistent system can be used to prove its own consistency.
I'm busy with proof theory and related pages, and don't want to apply these changes.  Any takers? ---- Charles Stewart 08:38, 22 Sep 2004 (UTC)
thread_title: [[Self-verifying systems]]

message: Last edit  deleted the following link:
:The confusions of Gödel, an analysis of the multiple problems and unsound foundations involved in Gödel's theorems (in four parts)''
I think the deleted text is bad POV, but contains useful information.  I've not time now to work out a better text, but hoefully I will have time on Wednesday if noone else has. ---- Charles Stewart 16:59, 25 Sep 2004 (UTC)
thread_title:  Bad/useless/uninformative link? 

message: It seems to me this entry seriously needs to lose the second person, mostly under Meaning of Gödel's theorems.  It seems a little out of place in a mathemetical article such as this.
--WayneMokane 07:21, 19 Dec 2004 (UTC)
thread_title:  Tone 

message: On the one hand, the article states 
::"What Gödel showed is that in most cases, such as in number theory or real analysis, you can never discover the complete list of axioms."
On the other hand, it also states 
::"or example both the real numbers and complex numbers have complete axiomatizations."
Can someone help me resolve these apparently contradictory statements?  I was under the impression that the reals could not have a complete axiomatization, since they contain the natural numbers.  Lethe | Talk 08:28, Feb 15, 2005 (UTC)
:I assume that the reference to complete axiomatisation for reals is to something like Tarski's axiomatisation of elementary geometry.  The point is that languages based on the natural tend to need induction or something equivalent to be useful, while there are uses for languages based on the reals without such strong reasoning principles.  But the point is misleading: the axiomatisations for real/complex numbers are anything but complete for all the purposes mathematicians make of them.  The text should be changed.  I'll tackle it if noone else does in the next couple of days. ---- Charles Stewart 11:00, 15 Feb 2005 (UTC)
thread_title:  real numbers complete? 

message: It seems to me that this article is in need of a major overhaul. The sections "Meaning of Gödel's theorems" and "Discussion and Implications" in particular are rather imprecise if not outright incorrect. For example, in the section "Meaning of Gödel's theorems" we read
: Gödel's theorems are theorems in first-order logic, and must ultimately be understood in that context.
First of all, Gödel's theorems are about logi, not in first order logic. The most natural reading of "theorem in first order logic" is "logically provable sentence", which surely is not meant. Also, Gödel's theorem does apply to e.g. the type system of Principia Mathematica - Gödel originally proved his theorems w.r.t. a system P obtained from that of Principia Mathematica by adding the Peano postulates as primitives - and to the so-called second order arithmetic studied in reverse mathematics. 
We also read that 
: While an infinite list of axioms may sound strange, this is exactly what's used in the usual axioms for the natural numbers, the Peano axioms: the inductive axiom is in fact an axiom schema - it states that if zero has any property and the successor of any natural number has that property, all natural numbers have that property - it does not specify which property and the only way to say in first-order logic that this is true of all properties is to have an infinite number of statements.
This is of course true, but uninformative and potentially misleading. It suggests that quantification over arbitrary properties can be expressed in FOL using an infinite number of statements, which is of course false. It might also be pointed out that the non-categoricity of the first order axioms stems from the fact that induction is only required to hold for arithmetically definable properties.
Further, we read that
: Gödel's theorem has another interpretation in the language of computer science. In first-order logic, theorems are recursively enumerable: you can write a computer program that will eventually generate any valid proof. You can ask if they satisfy the stronger property of being recursive: can you write a computer program to definitively determine if a statement is true or false? Gödel's theorem says that in general you cannot.
This is misleading. The set of theorems being recursive is not about deciding whether a sentence is true or false but about deciding whether it's provable. The above could also be read as claiming that the set of logical validities of first order logic is not recursive. This follows from the negative solution to the Entscheidungsproblem, not directly from Gödel's theorems, although these are of course related.
And we also find that
: Gödel himself only proved a technically slightly weaker version of the above theorems; the first proof for the versions stated above was given by J. Barkley Rosser in 1936.
This is already covered in the section about the first incompleteness theorem.
Finally there is a paragraph stating that
: Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's program towards a universal mathematical formalism. The generally agreed upon stance is that the second theorem is what specifically dealt this blow. However some believe it was the first, and others believe that neither did.
This is already covered in the section about second incompleteness theorem in more detail.
In the section "Discussion and Implications" we find an innocous statement
: Therefore, in order to establish the consistency of a system S, one needs to utilize some other more powerful system T, but a proof in T is not completely convincing unless T's consistency has already been established without using S.
This is again incorrect. We need a different system, not necessarily stronger than S, as remarked in the section about Gentzen's theorem (which I rewrote as it was nearly unintelligible previously). Also, what is important here is not really formal provability per se, but the fact that to prove the consistency of S we need to utilize some mathematical principles not formalized by S. Of course, this also means that if we want a formal theory in which the consistency of S is provable, it needs to formalize mathematical principles not formalized by S, but the important thing is about actual provability, not about formal provability, which is after all merely a combinatorical property.
We are also told that
: There are some who hold that a statement that is unprovable within a deductive system may be quite provable in a metalanguage. And what cannot be proven in that metalanguage can likely be proven in a meta-metalanguage, recursively, ad infinitum, in principle. By invoking a sort of super Theory of Types with an axiom of Reducibility -- which by an inductive assumption applies to the entire stack of languages -- one may, for all practical purposes, overcome the obstacle of incompleteness.
Which is utterly incorrect. After we have added all finite types, we still get Gödelian incompleteness, which leads to the introduction of the type level omega, which leads to the introduction of type level omega+1, and continuing of omega*2, of omega^omega and so forth. The problem is that in order to formulate this as a formal theory, we need what are known as ordinal notation systems, which assign natural numbers to constructive ordinals. Armed with these notations we can assign notations to type levels. But the general question of whether a natural number is the notation of a constructive ordinal, or whether two natural numbers denote the same contstructive ordinal, is very complex (the set of constructive ordinal (notations) is in the complexity class Pi^1 in the analytical hierarchy). In order to get a formal theory with constructive ordinal notations assigned to the types, we thus need some recursive way of selecting natural numbers which are indeed notations of constructive ordinals. One natural way to do this is to use so called autonomous progressions, in which we start with, say, the omega first levels of the type hierarchy and for every natural number for which it is provable that it is the notation of a constructive ordinal, add the level corresponding to that notation, and iterate the procedure. But this procedure will not exhaust all constructive ordinals, and the result will be a formal system suspect to Gödelian incompleteness. Solomon Feferman has analysed the concept of predicative provability using such autonomous progressions and shown that the so-called Schütte-Feferman ordinal of predicativity, Gamma_0, is the first constructive ordinal which can't be shown to be such by means of predicative reasoning. Friedman has shown that (a finite form of) the Kruskal Tree Theorem can't be proved predicatively. 
In fact, one implication of Gödel's theorem - to me the philosophically most significant - is the following "inexhaustibility" phenomenon, first noted by Gödel himself (in his Gibbs lecture)
: There can't be a consistent formal theory T accepted as correct in which all informally provable mathematical statements are (formally) provable.
For assume that there were such a formal theory T. Since we accept T as correct, we certainly accept its consistency. But then we can deduce that the Gödel sentence of T is true. But T can't prove its Gödel sentence. Therefore T does not contain all informally provable mathematical statements.
This leads us to the study of proof theoretical reflection and transfinite progressions of theories. Given a T we accept as correct, we should also accept T + Con(T). In fact, we should accept the following (local) reflection principle (which is a statement scheme) Ref_T for T:
: If T |- P, then P
which states that for any sentence P, if T proves P, then P is in fact true. Thus acceptance of T leads to acceptance of T_1 = T + Ref_T, which leads to acceptance of T_2 = T_1 + Ref_{T_1} and so forth. Using ordinal notations we can extend this progression of theories first to T_omega = union of all T_i<omega, then to T_{omega+1} and so forth. This procedure is analogous to the one applied to the type hierarchy described above, and so if we are to find anything epistemologically relevant we must restrict the ordinal notation indices to such as can be proved to be notations of constructive ordinals in earlier theories in the progression. Such progressions have been studied first by Turing (under the name ordinal logics) and later by Solomon Feferman. (If we don't restrict the notations to such as can be proved to notations in earlier theories we can get a sequence of theories in which every true arithmetical sentence is provable. However, since the question of whether a natural number is the notation is of complexity Pi^1 (complete) this means that the result is of no epistemological interest whatsoever, since there is no reason to suppose mathematicians have the ability to tell of an arbitrary number whether it's a notation of a constructive ordinal or not. (Actually, we need a stronger principle known as the global reflection principle which says that if for all n, T |- P(n), then for all n, P(n).)). For more information about this kind of reflection and its philosophical significance, consult Torkel Franzen's book Inexhaustibility - a non-exhaustive treatment.
Reading furhter we find that
: The axiomatic system may consist of infinitely many axioms (as first-order Peano arithmetic does), but for Gödel's theorem to apply, there has to be an effective algorithm which is able to check proofs for correctness. For instance, one might take the set of all first-order sentences which are true in the standard model of the natural numbers. This system is complete; Gödel's theorem does not apply because there is no effective procedure that decides if a given sentence is an axiom. In fact, that this is so is a consequence of Gödel's first incompleteness theorem.
It is indeed a consequence of Gödel's first incompleteness theorem that the set of true arithmetical sentences is not recursively enumerable (and thus not recursive either). However, there are many non-recursive sets of axioms to which Gödel's theorem applies. For example, the set of all true Pi_1 sentences has a Gödel sentence. The crucial difference between e.g. the set of all true Pi_1 sentences and the set of all true sentences is that the former is arithmetically definable while the latter is not. (In fact this result of Tarski's follows from the generalization of Gödel's theorem, since if the set of all true arithmetical sentences were arithmetically definable, it would have a true Gödel sentence which would then be in the set, and it would be inconsistent. It appears that Gödel arrived at his incompleteness theorems by attempting to prove the consistency of analysis by constructing a truth definition and showing that the axioms are true.)
Also, the last three paragraphs seem entirely redundant.
In the section about misconceptions about Gödel's theorems it is stated that
: The theorem only applies to systems that allow you to define the natural numbers as a set. It is not sufficient that the system contain the natural numbers. You must also be able to express the concept "x is a natural number" using your axioms and first-order logic. There are plenty of systems that contain the natural numbers and are complete. For example both the real numbers and complex numbers have complete axiomatizations.
The intent here is laudable, but it's not quite clear what "define the natural numbers as a set", "You must also be able to express the concept "x is a natural number" using your axioms and first-order logic" or "the system contain the natural numbers" are to mean exactly. Perhaps a better formulation would be
: There are theories that are naturally interpreted in structures which contain natural numbers but to which Gödel's theorems do not apply. For example, the (elementary) theory T of real numbers is naturally interpreted in the structure (R,+,*,/,=,<). The reason Gödel's theorems do not apply to T is that even though the natural domain (R) contains the natural numbers there is no formula P(x) in the language of T, such that P(a) is true only of a is a natural number. Thus we can't express such concepts as "is a natural number coding a proof of contradiction from T" and the proof of Gödel's theorems do not go trough.
Also, it's entirely unclear to me what the following is supposed to tell us
: The theorem only applies to systems that are used as their own proof systems. Gentzen's work shows the consequences of using a proof theory that is not the same theory, but a more powerful one.
What does it mean to use a "system as its own proof system"? Also, "proof theory" usually refers to a discipline of mathematical logic, not to any formal theory or system and Gentzen's work seems to have nothing to do with using "more powerful proof theory", whatever that means. As I note in the section on Gentzen's theorem, it actually shows that the consistency of PA can be proved in a theory which is incomparable to PA w.r.t. strength. Thus this "correction" seems to me to engender just more confusion.
The proof sketch given in the aptly named section "Proof Sketch for the first theorem" follows, in my opinion, Gödel's original proof too slavishly. I think it's better to first note that all recursive functions are representable in formal systems "in which basic arithmetical facts are provable", prove the diagonal lemma and construct the Gödel sentence as the fixed point of the formula ~Prov_T(x).
I also think that the section on Gödel's second incompleteness theorem I wrote is sufficiently detailed to render the section "Proof sketch for the second theorem" redundant.
I wonder whether examples of undeciable decision problems really belong in this article, which is about another concept of undecidability. I added an explanation about this difference, but I still wonder whether listing such examples in an article about Gödel's incompleteness theorems is not merely confusing.
I've only picked the most egregious paragraphs for more or less detailed criticism. In addition to these, the article seems to lack coherence. As said, a major overhaul is needed. I'd like to rework and rewrite the offending sections from scratch, so if you have any objections or additional issues you wish to be addressed, please speak up.
Grr... I cannot make ωωω render correctly.  What gives?
--Aatu 12:29, 15 July 2005 (UTC)
thread_title:  A major overhaul needed?