# Questions tagged [nonstandard-analysis]

Non-standard ordered fields are fields which have infinitesimals, that is, positive numbers which are smaller than any positive *real* number. Non-standard analysis is analysis done over such fields (e.g. hyperreal fields). Please specify the exact framework for non-standard analysis you are using in your question (e.g., what definition of "hyperreal number" you are using).

457 questions
Filter by
Sorted by
Tagged with
105 views

### In what sense is $\Bbb R(x)$ an "instantiation" of the hyperreals?

I'm teaching myself about hyperreal numbers. My main motivation for doing so is that they include infinite numbers, whose existence I hear disputed & doubted often as "quantifying the ...
14 views

### How to make sense of the definition for the derivative using the standard part function?

If the standard part function of Δy/Δx is dy/dx. How do you show that using the increment theorem? Increment Theorem: Δy = dy + εΔx => Δy = f'(x)Δx + εΔx I tried it but I don't know how to go ...
58 views

### Transfinitely iterating the Puiseux, Levi-Civita, or Hahn series constructions

There are many ways to take some real-closed field and generate a proper extension of it with elements that are infinite and infinitesimal relative to the original field. One well-known example is ...
92 views

### Why does this nonstandard model of Robinson Arithmetic fail to be a model of PA?

As talked about in this question, there is a nonstandard model of Robinson arithmetic of the form: $z_0 + z_1\omega + z_2\omega^2 + z_3\omega^3 + ... + z_n\omega^n$ in which $\omega$ is a formal ...
43 views

### Levi-Civita field vs Puiseux series: why is Cauchy completeness important?

The Levi-Civita field's main claim to fame is that it's the smallest real-closed field which is a proper extension of the reals and which is Cauchy-complete. That last bit is important, or else the ...
1 vote
100 views

### "Real-closed" vs "transfer principle"

The hyperreals are "real-closed," which means that any first-order statement that is true of the reals is true of the hyperreals. However, they also satisfy a "transfer principle," ...
1 vote
49 views

### Formula for probabilistically selecting a good server

There are $N$ nodes, each has response time $t_i$ (here and further I use $i$ as an index). I want probabilistically choose a node with low response time; if there are several nodes with low response ...
360 views

97 views

### Keisler measures obtained from hyperfinite samples

Let $M$ be an infinite model. Let $F$ be the set of maps $M\to\mathbb R^+\cup\{0\}$ with finite support (i.e. 0 almost everywhere). Let $\langle M,\mathbb R, F\rangle$ a 3-sorted expansion of $M$ in a ...
68 views

### Can every nonarchimedean ordered field be embedded in some hyperreal field?

Let $F$ be a nonarchimedean ordered field. Is there always a hyperreal field $^*\mathbb{R}$ such that there is an embedding of $F$ in $^*\mathbb{R}$? As far as I understand it, the answer here implies ...
78 views

### In what way does the nonstandard definition of microcontinuity differ from that of epsilon-delta continuity, and related quibbles

During a very cursory glance over the Wikipedia articles on non-standard calculus, I spotted the following definitions of continuity, uniform continuity and "microcontinuity", and I wonder ...
79 views

66 views

### A positive number smaller than any positive infinitesimals

I've been searching everywhere for an answer, but to no avail. I know that the hyperreals are a quotient of the space $\mathbb{R}^\mathbb{N}$ of sequences of real numbers modulo a fixed free ...
34 views

### Is hyperreal field internally ismorphic to the real field?

According to this answer $^{*}\mathbb{R}$ is internally Dedekind complete. If I am not wrong, by uniqueness of $\mathbb{R}$, that makes $^{*}\mathbb{R}$ internally isomorphic to $\mathbb{R}$. But this ...
22 views

### Internal subsets of nonstandard extensions.

I am studying the first chapter of L. O. Arkeryd et. al: Nonstandard Analysis. Theory and Applications. There it is shown that for the multiset $(\mathbb{X}, \mathcal{P}(\mathbb{X}))$ it is possible ...
1 vote
60 views

### Are there any significant/meaningful ultrapowers other than the hyperreals?

I have recently begun reading about non-standard analysis. According to this wikipedia article it is possible to construct an ultrapower $M^I/\mathcal{U}$ from any structure $M$ and index set $I$ with ...