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33
votes
2answers
12k views

Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows: All numbers are divided into two classes: those ...
31
votes
7answers
5k views

Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...
8
votes
5answers
2k views

Interesting meta-meta-mathematical theorems?

The Goedel incompleteness theorems can be considered meta-mathematical theorems, as they are "written" in a meta-theory and "talk" about properties of a class of formal theories. The following may be ...
19
votes
6answers
2k views

Can a problem be simultaneously polynomial time and undecidable?

The Robertson-Seymour theorem on graph minors leads to some interesting conundrums. The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As ...
28
votes
15answers
5k views

Abstract Thought vs Calculation

Jeremy Avigad and Erich Reck in their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory" claim that one of the factors of becoming ...
13
votes
2answers
2k views

Clarification of Gödel's second incompleteness theorem

I am sorry for the following question, because the actual answer to this question is in the beautiful works of Feferman and Jeroslow, but, unfortunately, I havn't any time to go into that specific ...
70
votes
16answers
16k views

What if Current Foundations of Mathematics are Inconsistent? [closed]

The title of the question is also the title of a talk by Vladimir Voevodsky, available here. Had this kind of opinion been expressed before? EDIT. Thanks to all answerers, commentators, voters, ...
48
votes
12answers
9k views

Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic. The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...
42
votes
9answers
4k views

How do they verify a verifier of formalized proofs?

In an unrelated thread Sam Nead intrigued me by mentioning a formalized proof of the Jordan curve theorem. I then found that there are at least two, made on two different systems. This is quite an ...
50
votes
3answers
6k views

Nelson's program to show inconsistency of ZF

At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say: Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...
15
votes
1answer
1k views

Martin's “Philosophical Issues about the Hierarchy of Sets”

Some months ago (October 2010), in the context of the Workshop on Set Theory and the Philosophy of Mathematics, Professor Donald A. Martin gave a talk entitled "Philosophical issues about the ...
14
votes
6answers
1k views

Extensional theorems mostly used intensionally

Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts: $$ \int_a^b f(x)g'(x)ds = ...
7
votes
3answers
1k views

randomness in nature [closed]

What is the explanation of the apparent randomness of high-level phenomena in nature? For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...
17
votes
8answers
1k views

The Importance of ZF

It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's ...
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 ...
8
votes
3answers
2k views

Use of Conjectures to Prove a Theorem

Name a theorem T that has a proof based upon the truth of a conjecture C, and also has another proof based upon the falsehood of the same conjecture C, but for longtime has no known direct proof that ...
6
votes
2answers
412 views

Can we define an “empirically generic” real number?

Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...