-4
votes
1answer
162 views

A general question on nonnegative integer sequence [closed]

Let $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x\ $ with some conditions$\ \}$. Let $B=\mathbb Z_{\ge 0}-A$. Define $\ 2A= \{a+b : a \in A,\ b \in A\}$. Define $\ 2B=\{a+b : a \in B,\ b \in B\}$. Then the set ...
4
votes
1answer
215 views

Circulant matrix with integer entries and determinant 1 or -1

CONJECTURE Let $A= (c_0,c_1,\ldots,c_n)$ be a circulant matrix, i.e if $(c_0,c_1,\ldots,c_n)$ is the first column of $A$ then the $i$th column of $A$ is obtained by applying the permutation ...
21
votes
2answers
1k views

Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ... On the other hand model theory, in particular after Hrushovski, found many ...
6
votes
2answers
433 views

The Theory of Transfinite Diophantine Equations [closed]

The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for ...
4
votes
4answers
628 views

Prime numbers and limit ordinals

As a set, i.e. as a von Neumann ordinal, the $\omega$-th limit ordinal $\omega^2$ is fairly complex and not so easy to visualize (for the novice). But as an explicit well-ordering of $\mathbb{N}$, ...
22
votes
1answer
642 views

Does an existence of large cardinals have implications in number theory or combinatorics?

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...
0
votes
0answers
103 views

Operating on a set of sequences - such as adding sequences and so on possible even when sequence is coded as number?

From http://math.stackexchange.com/questions/346680/operating-on-a-set-of-sequences-such-as-adding-sequences-and-so-on-possible-ev Suppose that there is a way to code some set of sequences into ...
1
vote
1answer
810 views

Are there non-commutative models of arithmetic which have a prime number structure?

Peano Arithmetic (PA) models have a prime number structure and commutativity of addition and multiplication. Presburger arithmetic (PrA) models of arithmetic have addition without multiplication and ...
-1
votes
1answer
547 views

Axiom of Choice and Number Theory [closed]

There are so many applications of the Axiom of Choice (and consequently its equivalents) in number theory. But do you know any application of the Zorn's Lemma in Number Theory !? I mean a theorem or ...
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 ...
10
votes
3answers
424 views

Applications of idempotent ultrafilters

Recently Justin Moore has posted a solution to the amenability of Thompson's group F. A key(?) step exploits the existence of idempotent ultrafilters on $\mathbb N$ to construct an idempotent measure ...
1
vote
1answer
335 views

Absoluteness of Countability

Let M be a countable transitive model for ZFC, P is a partial order in M. Notions like "partial orders" and "dense" are absolute. Consider the following set $S$={$D\in M: D$ is dense in $P$} = {$D: D$ ...
4
votes
2answers
293 views

Heights of several interesting posets

Let the height of a poset $P$ be the supremum of ordinals that are order types of all well-ordered subsets of $P$ (with order inherited from $P$). Define several sets of total functions, in each ...
12
votes
3answers
1k views

How to prove that every real number is a zero of some power series with rational coefficients (if true)

How would one approach proving that every real number is a zero of some power series with rational coefficients? I suspect that it is true, but there may exist some zero of a non-analytic function ...
5
votes
1answer
353 views

Does second-order arithmetic prove every expressible instance of Dependent Choice?

Let bigset(X) and rel(X,Y) be otherwise arbitrary formulas in the language of second-order arithmetic with the indicated variables free, and thmemberof(Z,x,X) be the formula asserting that X is the ...
48
votes
5answers
8k views

Inaccessible cardinals and Andrew Wiles's proof

In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals. Here's the link: ...
35
votes
7answers
2k views

How would one even begin to try to prove that a simple number-theoretic statement is undecidable?

This question is closely related to this one: Knuth's intuition that Goldbach might be unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other ...
81
votes
11answers
16k views

Knuth's intuition that Goldbach might be unprovable

Knuth's intuition that Goldbach's conjecture (every even number greater than 2 can be written as a sum of two primes) might be one of the statements that can neither be proved nor disproved really ...
4
votes
2answers
822 views

A question about fields of real numbers.

Assume that the Continuum Hypothesis holds. If F is an uncountable field of real numbers, does F always necessesarily contain a proper uncountable sub-field? Are there many specific uncountable fields ...