6
votes
4answers
697 views

Does there exist a non-trivial Ultrafinitist set theory?

Does there exist a set theory T-which has not yet been proved to be inconsistent-and in which one can prove the existence of (1) the empty set (2) sets that are singletons and (3) sets which have ...
7
votes
3answers
1k views

Unprovable sentence about integers

Is there any natural* statement S about the natural integers such that if PA contains no contradictions then neither PA+S nor PA+not S contains a contradiction? If unknown, where can I read about the ...
6
votes
5answers
2k views

A meta-mathematical question related to Hilbert tenth problem

I am reading Bjorn Poonen's very nice survey on Hilbert's Tenth problem (http://www-math.mit.edu/~poonen/papers/uniform.pdf), and while I believe I understand the mathematics well, I have widespread ...
13
votes
5answers
2k views

What is the “reason” for modularity results?

The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? . I ...
7
votes
7answers
2k views

Meaning of Kronecker's comment to Lindemann

At the Mactutor history page, it is said that Kronecker remarked to Lindemann: "What good your beautiful proof on [the transcendence of] π? Why investigate such problems, given that irrational ...
17
votes
8answers
4k views

To what extent is it true that “number theory = mathematics”? [closed]

In a thought-provoking answer to this MO question, Kevin Buzzard and several commentators have described a multitude of ways in which number theory is related to other parts of mathematics. It seems ...
6
votes
4answers
1k views

Alternative axiom to induction

Is anyone aware of alternative axioms to induction? To be precise, consider peano axioms without induction PA-. Is there any axiom/axiom schema that is equiconsistent to induction, assuming PA-? If ...
-3
votes
2answers
455 views

what is logic without a proof system [closed]

Is there a proof for no proof ?