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).

Filter by
Sorted by
Tagged with
0
votes
2answers
74 views

Does any theory of infinite quantities provide info for $-1$ raised to infinite power, its absolute value, finite part and a series or integral form?

I've heard about surreal numbers, hyperreal numbers, Hardy fields, nonstandard analysis, cardinal ariththmetic, ordinal arithmetic, games, etc. My impression is that neither of them can exactly show ...
1
vote
1answer
58 views

What is a starfinite set?

In the book Field Arithmetic by Fried and Jarden, the following definition is given on p. 273: Consider an enlargement of a higher order structure that contains both $P$ and $K$. Call the elements of ...
1
vote
1answer
49 views

Looking for a comprehensive text that compares the development of calculus using limits vs infinitesimals

I am looking for a book that covers the development of calculus using the ideas of limits working within the reals and also infinitesimals by extending the reals to hyperreals. I have seen people talk ...
1
vote
1answer
86 views

Thinking about a uniform probability distribution over the natural numbers

I well know that in standard probability theory there is no way to define a uniform probability distribution over the natural numbers in the strict sense, but nonetheless, I'm trying to form an ...
1
vote
0answers
25 views

$f:X\to Y$ is an open map. Fix $x\in X$. Why is $\bigcap_{x\in U\in T_1}{}^*f({}^*U)\subseteq {}^*f(\bigcap_{x\in U\in T_1}{}^*U)$ true?

Suppose that $f:X\to Y$ is an open map. Fix $x\in X$. Why is $\bigcap_{x\in U\in T_1}{}^*f({}^*U)\subseteq {}^*f(\bigcap_{x\in U\in T_1}{}^*U)$ true? https://iugspace.iugaza.edu.ps/bitstream/handle/20....
0
votes
0answers
28 views

$X,Y$ are topological spaces. Is that $f:X\to Y$ is open equivalent to $(\forall x\in X)(f^*(\mu(x))\supseteq \mu(f^*(x^*)))$?

I wanted to apply nonstandard analysis techniques to topology theory, but got trouble finding the equivalent condition to the openness. It would take too much time to explain all definitions from the ...
1
vote
2answers
47 views

Lectures on Non-Standard Analysis Book

I'm undergrad Applied Math Student and I'm writing my thesis about Non-Standard Analysis. I find a Spriger Book named Lectures On Non-Standard Analysis which are based on short courses of lectures ...
0
votes
0answers
16 views

Reference Request: Calculus on $\frac{R[x]}{(x^2)}$

Given a communative ring $R$, we can define a ring that I'll call $R^\epsilon$ by $$ R^\epsilon = \frac{R[x]}{(x^2)} $$ which we can easily think of as the ring generated by terms like $a + b\epsilon$ ...
2
votes
0answers
37 views

A hyperreal field is a minimal field such that …?

The standard presentation of hyperreals is difficult to understand. One typically motivates the hyperreals by a desire to have a consistent theory of infinitesimal elements, and then introduces the ...
0
votes
1answer
54 views

Existence of an inverse number to zero (null)? Existence a meta of number? [duplicate]

Sorry for the structure and maybe some mistakes in the post - I'm not a Mathematician and not a native English speaker, so I will use Google Translator + Grammarly for help. Please, if you want and ...
0
votes
0answers
34 views

A term that can be used to call infinite and finite quantities but not infinitesimals?

Suppose you operate with finite and infinite entities or numbers, and also with infinitesimals. Is there an established term that unites the first two but excludes infinitesimals? For instance, ...
2
votes
0answers
84 views

Ultraproduct construction: are finite hyperreals just a thinly disguised version of Cauchy sequences?

Periodically I've tried to wrap my head around nonstandard calculus and hyperreals, but I always thought I needed a lot more of a background in formal logic and/or set theory to understand what's ...
1
vote
1answer
66 views

Attempt to define limit of a sequence of surreal numbers

For sake of well-definedness, here we consider only ordinals less than the first uncountable ordinal, $\Omega$. Just like $\infty$ in the notation $\lim_{n→\infty}$ is essentially $\omega$, $\Omega$ ...
6
votes
1answer
137 views

division by prime numbers on non standard models

I am currently studying first-order logic and I am struggling on a problem. We work on a first-order language with non-logical symbols of arithmetics and the axioms of arithmetic. We define a non-...
1
vote
0answers
169 views

Probability theory on the transfinite

The usual fomalization of probability through $\sigma$-algebras and $\sigma$-additive measures can effectively model (countable) infinite chains of trials. This is usually done by defining a countable ...
1
vote
1answer
54 views

Quantifier Elimination for the theory of hyperreals with a much less than relation

We define a binary predicate $\ll$ over hyperreals as follows: $x \ll y$ if for every positive standard real number $r$, we have that $0 \le rx < y$. Now consider the first-order theory of true ...
0
votes
1answer
61 views

Extensions of branches to nonstandard trees

This is sort of inspired by the "cute" nonstandard proof of the fact that an infinite but finitely branched rooted tree has an infinite branch. An infinite branch is found by taking ...
2
votes
1answer
116 views

Distinction of hyperreals with sets

Let $^\ast\mathbb{R}$ be the set of hyperreals, constructed as a non-principal ultraproduct over the reals, and let $x\in {^\ast\mathbb{R}}$ and $y\in {^\ast\mathbb{R}}$ be two different hyperreal ...
0
votes
1answer
48 views

Why don’t sequences of real numbers with pointwise operations form a field?

I found this statement in a book introducing nonstandard analysis. It also says that “For example, let $E$ be the set of even natural numbers, and let $O$ be the set of odd natural numbers. The ...
2
votes
1answer
68 views

Mapping real coordinate space to hyperreal numbers while preserving “lexicographic order”

Coming up with a function $f:X^n \rightarrow \mathbb{R}$ where $X$ is a finite set of whole numbers such that lexicographic order is preserved is straightforward: $$f(x_1, x_2, \ldots ,x_{n-1}, x_n)=\...
1
vote
1answer
70 views

Status of the first infinite ordinal $\omega$ within non-standard analysis?

With newfound freetime during the pandemic, I have been studying non-standard analysis. I wasn't too fond of ultrafilters, so I've gravitated toward Nelson's internal set theory and Hrbacek set theory....
4
votes
1answer
182 views

Does 3Blue1Brown's series on Calculus : Essence of Calculus approach it via limits or infinitesimals (or both)?

I was introduced to Calculus by the online series on it by Grant Sanderson (3Blue1Brown's owner) called Essence of Calculus. In his videos, he treats $dx$ as $\Delta x$ that approaches $0$ and $dy$ as ...
2
votes
1answer
78 views

In the foundations of NSA with ultrapowers, how much can the axiom of choice be weakened?

The derivations I've seen of the hyperreals using ultrapowers use the axiom of choice and Zorn's lemma a lot. But looking closer, you can possibly weaken the axioms used in the derivations of some ...
1
vote
0answers
17 views

Nonstandard extension of nonstandard hull

Let $(X_i, d_i, e_i)$ be a sequence of pointed metric spaces, let $\prod _\omega (X_i, d_i, e_i)$ be the ultraproduct of said spaces with respect to a nonprincipal ultrafilter $\omega$, and let $(\hat{...
0
votes
0answers
89 views

Clarification on type of proof argument

I am wondering what it means to prove something via the saturation argument. I have seen this appear in a number of questions, but I am not sure as to what a saturation argument is. Any help would be ...
3
votes
0answers
88 views

Isomorphism of hyperreal fields viewed as extensions of the real field

Crossposted on MathOverflow: https://mathoverflow.net/q/368381/461 Let $A$ be the $\mathbb R$-algebra of all $\mathbb R$-valued functions on $\mathbb N$, that is $$ A=\mathbb R^{\mathbb N}=\prod_{n\in ...
1
vote
1answer
53 views

denominator with standard part 0

(i) Why do we have to simplify the numerator and denominator, and not just substitute the standard part st(c) right away since it is given as 4. (ii) Also what does this mean (more specifically, what ...
0
votes
2answers
86 views

What does it mean for an expression to be “finite but not infinitesimal”?

Also according to the definition of a positive infinitesimal: a hyperreal number b is positive infinitesimal if b is positive but less than every positive real number. So how are real numbers other ...
0
votes
0answers
25 views

Non-standard analysis bounded growth sequence problem

I am using non-standard analysis to solve a bounded growth problem. The problem: Let $ (a_n)_{n \ge 1} , a_n \in \mathbb{R}, $ a bounded growth sequence. Calculate: $$ \, \lim_{n \to \infty} {(2a_n - ...
6
votes
1answer
102 views

What is $\sin{ω}$?

I am reading about hyperreal numbers defined as (to my understanding) certain equivalence classes on all sequences of real numbers. $ω$ is defined as $(1, 2, 3, ...)$, and all functions are applied ...
3
votes
2answers
136 views

Are geometric arguments using infinitesimals valid?

This question pertains to smooth infinitesimal analysis as presented in the book A Primer of Infinitesimal Analysis by John Bell. The book uses intuitionistic logic. Let $\Delta$ denote the set of ...
2
votes
1answer
116 views

Hyperreals, other models and 1=0.999…

Please dont jump on me before reading it all. I am aware and l agree that Within the Standard Reals 1=0.999..... Now, I know only a bit about the Hyperreals and other non-standard models of the Reals....
1
vote
1answer
45 views

Hyperreal Numbers (Sequence Definition)

I am trying to understand the definition of a limit (for a sequence) regarding hyperreal numbers converging to $L$. The definition (see link here) states a real sequence of numbers converges to $L$ if ...
3
votes
0answers
137 views

Reference request: How is $0.99\cdots$ defined in nonstandard analysis?

In this answer to the question Is it true that $0.999999999\dots=1$?, Noah Snyder points out that Symbols don't mean anything in particular until you've defined what you mean by them. This ...
0
votes
2answers
102 views

What is the hyperreal multiplicative inverse of $1 + \epsilon$, and how do we show it exists?

What is the multiplicative inverse of $1 + \epsilon$, in the ordered field of hyperreals or surreals? Simple algebra shows it must be equal $1-\epsilon+\epsilon^2-\epsilon^3+\epsilon^4...$ But how do ...
3
votes
2answers
232 views

Internal set theory: proof that limited integers are standard

I'm following this pdf from Edward Nelson about internal set theory: https://web.math.princeton.edu/~nelson/books/1.pdf I'm at page 6. Only two axiom schemes have been introduced so far. The ...
2
votes
2answers
103 views

Application of the transfer theorem in elementary calculus (Davis' Applied nonstandard analysis)

In Davis' Applied nonstandard analysis a proof of the following, often seen, proposition is presented: For a sequence $S_n$ $S_n \rightarrow L$ iff $S_n \approxeq L$ for all infinite n. He then ...
2
votes
1answer
66 views

Can the Dedekind completion of the hyperreal numbers be embedded in an ordered field?

This answer shows that the Dedekind completion of the set of hyperreal numbers, endowed with the usual definitions of addition and multiplication of Dedekind cuts, is not an ordered field. But my ...
2
votes
1answer
45 views

Doubt in geometry of nilpotent elements.

I was reading commutative algebra from Miles Reid where I got stuck in the geometry of nilpotent element on page 29. The geometric picture of nilpotents is in the spirit of a nonrigorous ...
1
vote
0answers
32 views

Physical/Geometric models for Reals , of different Cardinality (Lowenheim-Skolem, etc)?

The Real line is a model for the Standard Reals. Are there also known geometric or other ways of representing models of the Reals of different cardinality, e.g., the Hyperreals?
2
votes
1answer
79 views

Rigorous definition of integration of hyperreal functions?

Motivated by this question, I am curious to see whether the Dirac delta function could be represented with a hyperreal function using the following "hyperreal Gaussian:" $$\delta(x)=\sqrt{\frac{H}{\...
1
vote
1answer
95 views

Hrbacek Paradox

Would it be possible to give a high level explanation of what is going on to give Hrbacek's Paradox (and why it is called a Paradox)? "No infinite internal set X can be well ordered nor does it have ...
-1
votes
1answer
52 views

Another question on the Hyperreals - regarding the monad at infinity…

I'm interested in exploring whether there is a monad at infinity. I guess we would define the infinitesimal space surrounding infinity as "A number that is greater than any Real number, but smaller ...
1
vote
1answer
48 views

Nonstandard part of a limited hyperreal

Let $b$ be a limited hyperreal and $x$ be its standard part, i.e. the unique real number infinitely close to $b$. Is it true that one can find an infinitesimal $\varepsilon$ such that $$b = \frac{x}{...
1
vote
1answer
72 views

Ordering the hyperreals and infinitesimals

I'm just getting into the hyperreals and infinitesimals and I would like to understand how one can determine when e is <, = or > g (where e and g are elements of the the infinitesimals). How does ...
2
votes
1answer
49 views

Well-definedness of pointwise addition of hyperreals

I'm reading An Introduction to Nonstandard Analysis by Isaac Davis, and I'm confused on one of the lemma inside which helps to prove that the set of hyperreals form a field. Note that $\mathcal{U}$ ...
2
votes
1answer
166 views

Can an infinite sum of a nonzero constant equal a finite number?

In the real number system, for example, the sum $\lim_{N \rightarrow \infty} \sum^N_{i=1} (\frac{1}{N}) = 1$, but the individual terms tend to zero due to the fact $\lim_{N \rightarrow \infty} \frac{1}...
1
vote
1answer
34 views

The inverse of a finite number greater than $1$ in absolute magnitude is finite. What about the inverse of a finite number less than $1$?

A hyperreal $\epsilon$ is infinitesimal if for every standard natural number $n$, either $-1/n < \epsilon$, or $\epsilon < 1/n$. (Here, our relationships are in the context of some ultrafilter ...
2
votes
3answers
53 views

Is a specific choice of ultrafilter necessary in order to get “concrete results” in nonstandard analysis?

Suppose we have sequences of real numbers which are indexed by the natural numbers. We can then define an ultrafilter $\mathcal{U} \subset 2^{\mathbb{N}}$ (where $2^{\mathbb{N}}$ is the powerset of ...
1
vote
1answer
55 views

Is a continuous net of hyperreal numbers eventually constant?

Let $(x_\alpha)$ be a net of hyperreal numbers indexed by the class of ordinals satisfying the condition that $x_\beta = {\lim}_{\alpha<\beta}(x_\alpha)$ for all limit ordinals $\beta$. Then my ...

1
2 3 4 5
9