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Questions tagged [arithmetic]

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1
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0answers
49 views

Imagine a universe with different laws of arithmetic. How would it be? [closed]

A universe with different mathematics and logic. Different laws of logic and mathematics (like different laws of arithmetic for example)
3
votes
3answers
197 views

Can we add to PA a new predicate T such that for every sentence A of the old vocabulary the new theory proves T(Godel numeric number of A) iff A

I am new to logic but I believe this is not a difficult problem, yet I am still soo confused, and the reason for that is because there are so many gaps in my knowledge or maybe I have overlooked so ...
8
votes
3answers
324 views

Can mathematical sentences in different theories be identified?

My question motivated by a part of this page from Saul Kripke's book Naming and Necessity, which is also viewable on google books. In the middle of the page he say something, which seems unnatural to ...
13
votes
5answers
1k views

How should we characterize the relationship between mathematics and philosophy of mathematics?

How should we characterize the relationship between mathematics and philosophy of mathematics? Specifically, in what ways might the study of philosophy of mathematics make a mathematician better at ...
16
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1answer
3k views

When it is correct to use Tarski's undefinability theorem versus Gödel's incompleteness theorem?

Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say ...
4
votes
2answers
474 views

Is there any possible world in which 2+2=5?

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, ...