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Gödel's incompleteness theorems show that arithmetic is either inconsistent or incomplete, and that arithmetic cannot prove its own consistency. It is useful to believe that arithmetic is consistent, and therefore also incomplete, but there are other points of view.

It seems to me that both views are compatible with the consistency of first-order logic itself. Thus, I am wondering: What is the autopsy of the efforts to revisit logic in the wake of Gödel? Is there any possible world in which 2+2=5, even if the shortest proof of it is gigabytes long, and individually we have no time to verify it without use of a computer? How do contemporary philosophers interpret Gödel as applied to this specific issue?

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    I am having a lot of trouble parsing exactly what it is you are asking. What do you mean by 'consistency' of first-order logic (FOL)? FOL is sound and complete for standard classical deductive systems (which strongly axiomatize it) - but I can't see what sense of 'consistency' you are talking about here? When you ask: "What is the autopsy of the efforts to revisit logic in the wake of Godel?" I have no idea what you mean. And I don't understand what the 'specific issue' refers to. I'm sure there's a very interesting question here - could you please clarify and explain? – Chuck Jun 11 '11 at 20:41
  • By consistency of FOL I just mean that we can't prove falsehood, so logic is not a science. That is a vacuous statement but it doesn't apply to arithmetic where other non-logical axioms are assumed. I was thinking of the development of intuitionistic logic and also abstract algebraic logic. Have any of the alternative or generalized logics developed in the last century had any impact on philosophy and can they help inform whether or not arithmetic is a science? – Dan Brumleve Jun 11 '11 at 21:01
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    There is some confusion here I feel - probably because of the admittedly confusing terminology. You have to bear in mind that when we speak of a 'logic' we have three things in mind: Language+Deductive System+Semantics. So Intuitionistic Logic (IL) is a first-order logic too if we take logic to be language+semantics - that is to say FOL=IL modulo language+semantics. IL differs with (what is commonly referred to as) FOL only in the choice of deductive system - e.g. in IL there is no RAA or double-negation elimination etc etc. Now are you asking: Did GT provide reasons to consider alternative – Chuck Jun 11 '11 at 22:02
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    (cont.) deductive systems for the underlying first-order logic of a theory of arithmetic? And if yes how have these systems affected philosophical perceptions of arithmetic? – Chuck Jun 11 '11 at 22:04
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    That is the broad part of the question; also I am asking about the specific thought experiment of having an actual proof of the inconsistency of PA which is too long or requires too much effort for a human to verify but can be verified easily by a computer. I can see there are a lot of alternatives here: conclude that PA is actually inconsistent, or that PA is actually consistent but there is a new law of physics causing the verification to falsely succeed, or even that no large computations can be trusted and all knowledge is finite, etc. What conclusion is the most plausible? – Dan Brumleve Jun 12 '11 at 0:01
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I think the question is a little unfocused. That said:

Gödel didn't show arithmetic inconsistent or incomplete. Rather, he showed that any formal system with expressive power adequate for expressing arithmetic was either inconsistent or incomplete and that no such system could prove its own consistency.

There are infinitely many proofs that 2+2=5 and they can be as short as you like. But, unless there really is an undetected inconsistency in a system of arithmetic (say First Order Peano Arithmetic), they will all be proofs in a system of "arithmetic" that has either false axioms or unsound rules of inference. I don't know what a possible world in which 2+2=5 would be like. It is epistemically possible that 2+2=5 is provable in FOPA, but, since FOPA also can prove 2+2=4 and ~(4=5), that would show FOPA inconsistent.

Gentzen did prove the consistency of Peano Arithmetic, but his proof relies upon mathematical induction up to epsilon-0, and is thus no use in persuading those sceptical of the consistency of PA. It is, however, an important contribution to proof theory, setting an upper bound on the principles on needs to show PA consistent. (By Gödel's second incompleteness theorem, that bound must be higher than the system of PA itself.)

  • In the inconsistent world the negation of the conjunction of the Peano axioms would be a tautology. Without induction as a logical inference rule, the same applies to the finite axioms of Robinson arithmetic. Presburger arithmetic is a justification for ~(2+2=5) but it doesn't justify 2*13=26 because "*" is not part of the language. I'm interested in understanding what such an inconsistent world would be like or what would be so absurd about it. – Dan Brumleve Jun 11 '11 at 6:18
  • How can an axiom be false? – Ben Hocking Jun 11 '11 at 11:47
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    Formally, you can make any sentence in the language of a system as an axiom. In a formal proof system, an axiom is simply a sentence which can be written down at any point in a proof with no further justification than that it is an axiom of the system. Of course, one typically doesn't want to have adopted falsehoods as axioms. It can, and does, happen, though. Frege's Basic Law V (which was an axiom) in his Grundgesetze system was shown, by Russell's Paradox, to be false. Likewise, Euclid's Firth Postulate would seem to be false if we take the axioms to characterize physical space. – vanden Jun 11 '11 at 15:54
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Godels theorems are about the expressive power of formal languages; and only tangentially touches on the nature of truth; for mathematicians, it's importance is in prompting the growth of a new field: model theory.

Though we don't have a world where 2+2=5; though if we did, we should ask what does it mean? Is it also the case that 2+2=4? So that 4=5? And what would this mean ... ?

Though we do not have a world where 2+2=5; we do already have a world in 2+1=0 and 2+1=3 - this one - it's a world where we have the cyclic group of order three; and many others besides.

The importance of non-Euclidean geometry is generally noted in contrast to that of the exemplary geometry - Euclidean geometry; and this by negating the parallel axiom.

What is not noted, is the importance of non-arithmetic arithmetics - groups, rings, fields; they are as usefully important as the usual, exemplary one; to give it a name - Peano Arithemetic.

Mathematical logic as opposed to logic has moved on; logic as Heidegger points out:

has the status of a secure and trustworthy science; it has taught the same thing since antiquity.

New concepts in mathematical logic would include, say, Topos theory; that demonstrates Cohens forcing technique as a species of geometry.

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