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11
votes
2answers
545 views

Inconsistency and workaday independence.

Set-theoretic topologists, for example, encounter many propositions that turn out independent from set theory. Sometimes these results require novel forcing arguments, but often they simply rely on ...
22
votes
7answers
2k views

Supervenience in mathematics

I'm not quite sure if this is the right place to ask, and if this is the right way to ask, but I dare. In philosophy (of mind, e.g.) the concept of supervenience is used: "Supervenience [is] used ...
27
votes
3answers
3k views

The influence of string theory on mathematics for philosophers.

I've agreed, perhaps unwisely, to give a talk to Philosophers about string theory. I'd like to give the philosophers an overview of the status and influence of string theory in physics, which I feel ...
9
votes
5answers
2k views

Proper classes and their consequences

I have two main questions: What is a proper class? I've read that it's collection of objects that's "too big" to be a set, but in what sense is such a collection "too big"? Since I'd like this post ...
23
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 ...
18
votes
2answers
2k views

Should there be a true model of set theory?

As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
14
votes
1answer
693 views

How is Fredkin and Toffoli's Conservative Logic related to Linear Logic?

In the answers to this question, Timothy Gowers asks: I've been interested in this question for some time. I haven't put any serious thought into it, so all I can offer is a further question ...
53
votes
9answers
6k views

Is there any formal foundation to ultrafinitism?

Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to wikipedia, it has been ...
45
votes
2answers
14k 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 ...
34
votes
7answers
6k 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 ...
12
votes
3answers
2k views

What was Weierstrass's counterexample to the Dirichlet Principle?

Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and ...
15
votes
2answers
3k views

Question arising from Voevodsky's talk on inconsistency

This question arises from the talk by Voevodsky mentioned in this recent MO question. On one of his slides, Voevodsky says that a general formula even with one free variable describes a subset of ...
74
votes
16answers
18k 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, ...
16
votes
2answers
2k views

Universe view vs. Multiverse view of Set Theory

Here I refer to Hamkins' slides: http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf particularly, to the "Universe view simulated inside Multiverse", p. 22. My question is: is it very unsound ...
12
votes
1answer
808 views

evil properties, higher category theory and well-chosen tensor products

Let's start with the following random example: If $F$ is a presheaf, then for every chain of open subsets $U \subseteq V \subseteq W$, the morphisms $F(W) \to F(V) \to F(U)$ and $F(W) \to F(U)$ ...
20
votes
6answers
2k views

Interpretation of the Second Incompleteness Theorem

For simplicity, let me pick a particular instance of G\"odel's Second Incompleteness Theorem: ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does ...
50
votes
3answers
7k 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 ...
17
votes
7answers
4k views

What is Realistic Mathematics?

This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers ...
10
votes
2answers
1k views

Proving Independence of Axioms by Exhibiting Models Which Don't Satisfy Our Intuition

I recently saw the proof of the independence of ZF (with allowance for multiple empty sets) and AC. The proof constructed the model based on a set theory generated by infinitely many empty sets and ...
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 ...
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
982 views

When is a statement provable?

We all know that Gödel showed that there, in a formal system, are true statements that are non-provable (undecidable). In ZFC, there's Kaplansky's Conjecture, the Whitehead problem, etc. We can also ...
12
votes
3answers
1k views

Are there natural examples of mathematical statements which follow from consistency statements?

Motivation One of the methods for strictly extending a theory $T$ (which is axiomatizable and consistent, and includes enough arithmetic) is adding the sentence expressing the consistency of $T$ ( ...
64
votes
18answers
8k views

Can a mathematical definition be wrong?

This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...
13
votes
3answers
1k views

Has there ever been a weaker Church-like thesis?

Background. The Church-Turing thesis, in one of its many equivalent formulations, states that the intuitively computable arithmetical functions are exactly those computed by Turing machines. ...
31
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 ...
10
votes
6answers
1k views

Defining variable, symbol, indeterminate and parameter

Are there precise definitions for what a variable, a symbol, a name, an indeterminate, a meta-variable, and a parameter are? In informal mathematics, they are used in a variety of ways, and often in ...
19
votes
9answers
5k views

Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]

As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...
1
vote
2answers
734 views

Mathematics of the Anthropic Principle [closed]

A form of the anthropic principle is as follows: "We can observe the universe only because we can exist within it in some way such that we can observe it, and it exists such that we can observe it." ...
46
votes
34answers
10k views

nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because ...
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 ...
4
votes
0answers
625 views

BGG category everywhere implies generalized Kazhdan-Lusztig formula?

Maybe this question is vague. I am not an expert on what I asked, if I made mistake, please point out. BGG category was discoverd in Lie algebra setting. One has Verma module $M(\lambda)$, ...
18
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
1answer
859 views

Using the multiverse approach to decide the law of the exluded middle?

Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...
30
votes
5answers
4k views

Why do categorical foundationalists want to escape set theory?

This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full. I know that it's possible to ...
7
votes
3answers
822 views

What are trig classes like within a universe that's “noticeably” hyperbolic?

[I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.] What are trig classes like within a universe that's "noticeably"[*] ...
0
votes
1answer
530 views

Formal definition of 'useful' ?

Has anyone worked out a formal, general-enough definition of what is 'useful', so that it could reflectively be used in mathematics? I am aware of the work in utility theory from economics (but ...
18
votes
7answers
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 = ...
43
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 ...
2
votes
3answers
1k views

The problem of infinity [closed]

Background and motivation The following is copied from my blog since someone thought it was the clearest statement I had made regarding a problem I recently posed. On their advice, it is a community ...
-3
votes
2answers
1k views

Finite versus infinite on non-Hausdorff topologies [closed]

Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a ...
6
votes
5answers
3k views

Models of ZFC Set Theory - Getting Started

For just any first-order theory: What are the sets I am supposed/allowed to think of when thinking of models as sets (of something + additional structure)? Provided: I can think of models of any ...
1
vote
4answers
3k views

Are all mathematical theorems necessarily true?

Define a formal tautology as a statement where by the nature of its atomic components there exists no truth-value assignment where it is not true. A contingent statement is a statement that is true by ...
44
votes
2answers
5k views

Lawvere's “Some thoughts on the future of category theory.”

In Lecture Notes in Mathematics 1488, Lawvere writes the introduction to the Proceedings for a 1990 conference in Como. In this article, Lawvere, the inventor of Toposes and Algebraic Theories, ...
5
votes
1answer
901 views

Co-Objects are better [closed]

This is a rather vague question, but perhaps we can talk about it. There are two types of mathematical objects (which don't exclude each other): A) There is a good description of morphisms defined ...
0
votes
2answers
417 views

In what sense Fraissean view point shows Model Theory can be done without any formal syntax and deduction rule?

In this post I want to look at an issue I was in doubt when looking at the comment of F. G. Dorais in the post In model theory, does compactness easily imply completeness? F. G. Dorais remark was: ...
11
votes
2answers
807 views

Where are we working when we prove metamathematical theorems?

I am posting my comment from this question as a separate question, as was recommended to me. (EDIT: I'm sorry if it ended up being too similar a question, I just wanted to phrase it in the ...
32
votes
5answers
3k views

Were Bourbaki committed to set-theoretical reductionism?

A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent ...
14
votes
10answers
1k views

Can you prove equivalence without being able to calculate it?

In mathematics we often seek to classify objects up to an equivalence relation, where two objects A and B are said to be equivalent if there exists a map $f:A\rightarrow B$ satisfying certain ...
7
votes
3answers
2k 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 ...