# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?