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).
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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$f:X\to Y$ 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....
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$X,Y$ 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 ...
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2answers
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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 ...
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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 $a + b\epsilon$ ...
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0answers
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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 ...
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1answer
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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 ...
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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, ...
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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 ...
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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 $\lim_{nā\infty}$ is essentially $\omega$, $\Omega$ ...
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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-...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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1answer
48 views
Why donāt sequences of real numbers with pointwise operations form a field?
I found this statement in a book introducing nonstandard analysis. It also says that āFor example, let $E$ be the set of even natural numbers, and let $O$ be the set of odd natural numbers. The ...
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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)=\...
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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....
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1answer
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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 ...
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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 ...
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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{...
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0answers
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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 ...
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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 ...
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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 ...
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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 ...
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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 - ...
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1answer
102 views
What is $\sin{Ļ}$?
I am reading about hyperreal numbers defined as (to my understanding) certain equivalence classes on all sequences of real numbers. $Ļ$ is defined as $(1, 2, 3, ...)$, and all functions are applied ...
3
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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....
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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 ...
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0answers
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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 ...
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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 ...
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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
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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
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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
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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
...
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0answers
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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
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1answer
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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
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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 ...
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1answer
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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 ...
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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}{...
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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
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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
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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
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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 ...
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3answers
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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 ...
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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 ...
