The foundations tag has no usage guidance.

**4**

votes

**3**answers

2k views

### Dedekind's theorem

In "Was sind und was sollen die Zahlen?" Dedekind gives a noncircular
proof of the statement that a set is finite if and only if it cannot be
put in bijective correspondence with a proper subset. By ...

**1**

vote

**1**answer

258 views

### continuous maps between categories that are not functors

Hey,
Is it possible to define a map between two categories which preserves all products and binary equalizers and yet is not a functor, ie it does not satisfy one or more axioms of a functor? ...

**6**

votes

**2**answers

576 views

### Sets as Combinatorial Games

Just a few days ago my seemingly eternal and recurrent fascination for Conway's combinatorial game theory (CGT) & surreal numbers had a recrudescence, so I grabbed this excellent survey, and began ...

**10**

votes

**10**answers

4k views

### Is PA consistent? do we know it?

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs ...

**7**

votes

**3**answers

2k views

### incompleteness in real analysis

Godel's theorem tells us that any sufficiently powerful consistent formal theory of the integers is incomplete; but what about formal theories of the real numbers? More precisely, what about theories ...

**15**

votes

**1**answer

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 ...

**9**

votes

**6**answers

938 views

### (Non?)-linearity of the consistency strength ordering in ZF

Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all ...

**25**

votes

**6**answers

5k views

### How true are theorems proved by Coq?

Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to ...

**43**

votes

**7**answers

5k views

### Why hasn't mereology succeeded as an alternative to set theory?

I have recently run into this wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set ...

**8**

votes

**3**answers

1k views

### Kunen's use of Countable Transitive Models

Hi,
I have a doubt concerning Kunen's exposition of forcing in his classical book (arguably $the$ book on forcing). When dealing with Countable Transitive Models to set up the forcing machinery, ...

**9**

votes

**2**answers

617 views

### Consistent hierarchy of axiomatic systems

First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight.
I just learned in Sergey Melikhov's answer to another question ...

**5**

votes

**5**answers

1k views

### Concrete models of abstract structures

Most mathematicians seem to be contented with the fact, that abstract structures cannot be directly modelled as such in a set theory without ur-elements. What seems to me the standard stance: Set ...

**4**

votes

**4**answers

1k views

### Subsystems of Peano arithmetic and incompleteness theorem

I think everyone is familiar with Goedel's incompleteness theorems. In particular they imply that PA (Peano arithmetic) can not prove its own consistency. Now my question is what is the largest ...

**20**

votes

**1**answer

1k views

### Nontrivial circular arguments?

There is a famous circular argument for the Prime Number Theorem (PNT). It turns
out that there exists an infinite sequence of elementary-to-prove Chebyshev-type estimates
that taken together imply ...

**2**

votes

**0**answers

147 views

### Multitype approaches to choice?

I wonder if anyone has developed a set theory which approaches the issue of the non-emptiness of products of non-empty sets via a hierarchy of types (comparable to how Von Neumann–Bernays–Gödel set ...