3
votes
1answer
63 views

Ultrapower and hyperreals

The construction I've seen of the field of hyperreal numbers considers a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, then takes the quotient of $\mathbb{R}^{\mathbb{N}}$ by equivalence ...
1
vote
1answer
56 views

Saturation, (Complete) Ordered Fields and Model-Theoretic Methods in relation to Real & Non-Standard Analysis

I am trying to understand the following three questions: One and Two and Three. I'm under the impression that they're interrelated, though maybe not directly. What do I need to read to back-fill to ...
3
votes
1answer
46 views

Extended reals from ultraproduct of algebraic numbers

Let $\mathbb{A}$ denote the field of real algebraic numbers. Let $\mathcal U$ denote a free ultrafilter. Construct $F=\prod_{\mathcal U} \mathbb{A}$. This is a field containing $\mathbb A$, and we ...
2
votes
0answers
32 views

Given a Hardy Field is it always possible to find a smooth representative of each germ?

In this case I refer to a Hardy Field (of germs at infinity) $\mathcal{H}$ a a field of germs of real valued functions on $\mathbb{R}$ that is closed under differentiation. That is, if ...
2
votes
3answers
97 views

Prove $\forall r \in \mathbb{R}. \exists k \in \mathbb{Z}. r < k$

I would like to prove that for every real number there exists an integer that is greater than it. My problem lies in that I am not sure how to construct the real numbers and provide their theory with ...
24
votes
2answers
445 views

Is $ \pi $ definable in $(\Bbb R,0,1,+,×, <,\exp) $?

Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields together with the unary function symbol $ \exp $ such that in the standard ...
4
votes
1answer
99 views

What is the formulation of the Least Upper Bound propierty in First Order Logic?

I've been readining about the completeness Godel's theorems. Accordingly, the axioms of $R$ in first order logic make up one of these sets that is complete and consistent. But I've always seen the ...
2
votes
2answers
146 views

how to create the set of hyperreal numbers using ultraproduct

As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?
14
votes
2answers
286 views

Applications of model theory to analysis

Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from ...
2
votes
1answer
334 views

Formalising real numbers in set theory

If I understand correctly, the real numbers can be formalised in a first-order, like in ZF. However, such a formalisation is not strong enough to ensure that all models of the reals are isomorphic. ...