I've read an article, and I would like to ask the author for clarification, but I couldn't find contact details.

Link: http://www.mathpages.com/home/kmath347/kmath347.htm

Oddly enough, some people (even some mathematicians) are under the impression that Goedel’s results apply only to very limited formal systems. One mathematician wrote to me that “there is no proof in first-order logic that arithmetic is consistent, but that has more to do with the limitations of first-order logic than anything else, and there are other more general types of logic in which proofs of the consistency of arithmetic are available.” Of course, contrary to this individual’s claim, Goedel's results actually apply to any formal system that is sufficiently complex to encompass and be modeled by arithmetic. Granted, if we postulate a system that cannot be modeled (encoded) by arithmetic then other things are possible, but the consistency of such a system would be at least as doubtful as the consistency of the system we were trying to prove.

From my understanding, Gödel's theorems apply to formal systems that encompass, i.e. model arithmetic. What does the author mean when he says that a system is "modeled by" arithmetic?


1 Answer 1


Looking at the rest of the linked article, it seems that the author says that a system $T$ is "modeled by arithmetic" if the consistency of Peano Arithmetic implies the consistency of $T$ (or maybe the stronger claim that the system $T$ can be interpreted ("formalized") in Peano Arithmetic.) For example,

Of course, people sometimes raise the possibility of a finitistic system that cannot be modeled within the theory of natural numbers but, as Ernst Nagel remarked, "no one today appears to have a clear idea of what a finitistic proof would be like that is NOT capable of formulation within arithmetic".

PA can be modeled within ZF. It follows that con(ZF) implies con(PA).

The article itself is something of a rant, however - the author seems to carefully hide whatever thesis he is trying to make. Most of the content seems to be "correct", but I get the sense that some conclusion is left unstated.

A key question that the author seems not to have discussed is why mathematicians find the consistency proofs of Peano Arithmetic to be interesting even though they are not absolute consistency proofs.

  • $\begingroup$ Why do you mean by "absolute consistency proofs"? $\endgroup$ Mar 10, 2015 at 14:04

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