5
votes
2answers
478 views

There is no smallest infinitely large prime

I'm reading Kunen's Foundations of Mathematics and trying to solve Exercise II.16.19, which constructs an elementary extension of the reals as an ordered field, and asks the reader to prove various ...
3
votes
2answers
96 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 ...
4
votes
1answer
220 views

Construction of *ZFC

In the following paper, page 11 (Appendix), there is a construction of a model of a theory $^*ZFC$ (see the definitions in the paper included) from a model of $ZFC$. I have been trying really hard to ...
2
votes
1answer
70 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 ...
5
votes
2answers
88 views

define the reals in a non-archimedean elementary extension of the real field.

Can it be done? We have the real field $(\Bbb R,+,-,\times,0,1,<)$, of course $(0,1,-,<)$ are definable using the rest. We take an elementary non-archimedean extension. Can we define the ...
2
votes
0answers
43 views

Concurrent relation and enlargement

The superstructure $V({}^{\ast}X)$ [with respect to a monomophism $\ast : V(X) \to V({}^{\ast}X)$] is called an enlargement of $V(X) $ if for each set $A \in V(X)$ there is a set $B \in {}^{\ast} ...
0
votes
1answer
94 views

Question on the ultrapower construction of superstructure

On page 83-85 (here, on googlebooks), An Introduction to Nonstandard Real Analysis, Albert E. Hurd, Peter A. Loeb, two steps are given to construct a monomorphism $\ast : V(X) \to V({}^{\ast} X)$. The ...
3
votes
1answer
85 views

What's the idea behind the proof of saturation of internal sets via ultrapower construction?

I'm trying to understand the proof of saturation of internal sets via ultrapower construction on Robert Goldblatt's Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Though, it's ...
10
votes
1answer
118 views

Is there a universal property for the ultraproduct?

Given an ultrafilter U on a set I and a collection of ser X_i ($I \in I$) one defines the ultraproduct as the quotient of $\prod X_i $ by the identification $x_i=y_i :\leftrightarrow \{i:x_i=y_i\} \in ...
2
votes
1answer
108 views

Nonstandard structure of Presburger arithmetic

Let $\mathfrak {R}_A = (\Bbb {N}; 0, S,<,+)$. What can we say about ${}^{\ast}\Bbb N$, the universe of non-standard structure of the first order theory of $\mathfrak {R}_A$? Firstly, because of ...
6
votes
2answers
257 views

Why continuum hypothesis implies the unique hyperreal system, ${}^{\ast}{\Bbb R}$?

On page 33, Robert Goldblatt, Lectures on Hyperreals(1998): Now it has been shown under certain set-theoretic assumption called continuum hypothesis the choice of $\mathcal F$ is irrelevant: All ...
5
votes
2answers
77 views

How to show $\chi_{{}^{*}P} ={}^{*}\chi_{P}$ by transfer principle?

Let $\mathfrak{R}$ be the real number system, $(\mathbb{R},+,\cdot,<)$ and ${}^{*}\mathfrak{R}$ be the hyperreal number system $({}^{*}\mathbb{R},{}^{*}+,{}^{*}\cdot,{}^{*}<)$. Transfer ...
3
votes
1answer
173 views

Why every member of ${}^{*}\mathbb{R}$ is infinitely close to some member of ${}^{*}\mathbb{Q}$

Let $\mathbb{Q}$ be the set of rational numbers. Show that every member of ${}^{*}\mathbb{R}$ is infinitely close to some member of ${}^{*}\mathbb{Q}$. This is an exercise on page 180, A ...
2
votes
2answers
171 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?
5
votes
2answers
131 views

Nonstandard extension of a function with a limit

Question 1. Let $g : \mathbb{R} \to \mathbb{C}$ with $g(y) = \lim_{x \to \infty} f(x,y)$, where $f : \mathbb{R}^2 \to \mathbb{C}$. Is it correct that the nonstandard extension $^*g$ will have $x \in ...
5
votes
3answers
134 views

Superstructure with sets as atomic/base entities

This question arose while learning nonstandard analysis. The superstructure $V(X)$ of a nonempty set $X$ is defined recursively: $$\begin{align*}V_0(X) &= X \\ V_{i+1}(X) &= V_i(X) \cup ...
14
votes
1answer
509 views

Non-ZFC set theory and the hyperreals: problem solved?

The reals are the unique complete ordered field. The hyperreals $\mathbb{R}^*$ are not unique in ZFC, and many people seemed to think this was a serious objection to them. Abraham Robinson responded ...