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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Is there an external reflection principle in IST?
Background
The reflection principle is a theorem schema in ZF.
Given a formula $ φ $ and a set $ M $ we obtain $ φ $ relativized to $ M $ by restricting all quantifiers of $ φ $ to range over $ M $. ...
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Does this basic property of hyperreal function hold?
Let $x=(a_1, a_2, a_3, ...) + \mathcal U \in {}^\ast \mathbb R := \displaystyle\prod_1^\infty \mathbb R/\mathcal U$ be a hyperreal number using the ultrapower construction and $f \colon \mathbb R\to \...
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How to compare two hyperreals?
Potentially scattered question, this is the result of my trying and failing to understand the relevant wikipedia pages.
I know that every sequence of reals represents a hyperreal, but every hyperreal ...
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A rigorous treatment of the hyperreal numbers and nonstandard analysis.
I tried finding a decent book to study nonstandard analysis from and found Goldblatt's Lectures on the Hyperreals. However, I was very disappointed to find out that the text is not rigorous at all --- ...
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Largest subfields common to all hyperreal fields
Suppose we build the hyperreal numbers $^*\Bbb R$ in the usual way as an ultrapower of the reals, built as a quotient of $\omega$-length sequences of reals mod some non-principal ultrafilter. Then, in ...
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Does any infinite set contain an element that is not uniquely definable?
Background/Motivation
My framework is first order logic and ZF.
A uniquely definable set is one uniquely satisfying a predicate, for instance, the empty set is uniquely defined by the unary predicate $...
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How sensible is the limit from above against infinity in nonstandard analysis?
In nonstandard analysis, we have hyperreal numbers that are greater than any real number. As such, we can create a sequence of infinite, hyperfinite hyperreal numbers which grows ever smaller.
More ...
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Nonstandard Complex Analysis?
I recently discovered Nonstandard Analysis and am slowly working my way through Kelsier's textbook and Foundations companion. However while I have found plenty of stuff about real nonstandard ...
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How to find infinitesimally small segments for graph $y = x^2$?
I read on Wiki:
Intuitively, smooth infinitesimal analysis can be interpreted as
describing a world in which lines are made out of infinitesimally
small segments, not out of points.
I tried to find ...
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Uniqueness of standardization
So I'm reading Kanovei's and Reeken's book "Nonstandard Analysis, Axiomatically" and there is something which I'm not quite understanding.
So, in page 15, it is stated that a "...
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How to prove that 2 points $\varepsilon$ and 2$\varepsilon$ are indistinguishable not only in sole case? (Smooth infinitesimal analysis)
There are 2 points on the Real line: $A$ and $2A$. They are indistinguishable in sole case - if $A$ is $0$.
But how to prove that $\varepsilon$ and 2$\varepsilon$ are indistinguishable not only in ...
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Discontinuous function in Smooth infinitesimal analysis
I read that there isn't discontinuous functions in Smooth infinitesimal analysis. But I tried to define discontinuous function ($\varepsilon$ is infinitesimal):
$f(x) =
\begin{cases}
1, & \text{...
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Nonstandard analysis and hyperreals
Hyperreal numbers and nonstandard analysis are intimately connected. First, the nonstandard analysis consists of an extension of the axioms for the real numbers, which introduces a new predicate $\...
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Proposal for the number next to $0$ [duplicate]
What if we define the number next to $0$ on the real number line to be special like $0$, but not quite $0$. Is there some work done in this direction? Can someone point me towards it because I am ...
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standard part definition of the derivative
In high school my teacher told us : $$f'(a)= \lim_{h\to 0} \frac{f(a+h)-f(a)}{h} \tag{1}$$
For a long time I've been thinking that it was an alternative version of the other definition ($\lim_{x\to a}...
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Arguments illustrating advantage of hyperreal definition over sequential one
As is well known, fields of hyperreals $\mathbb R^*$ can be formed by
an ultrapower construction, as quotients of the space of sequences of
real numbers by a nonprincipal ultrafilter. In fact, some ...
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Nonstandard characterization of compactness
I'm trying to write up notes for myself on nonstandard analysis, consulting outside resources as little as possible. I have stumbled upon what seems to be referred to as Robinson's characterization of ...
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Why isn't Riemann sum infinitesimal in nonstandard analysis?
I am trying to learn nonstandard analysis from Keisler's book. In the integration chapter it feels like the use of the Transfer Principle is some kind of magic that just requires us to believe results ...
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Characterizing the set of first-order definable real functions
I am trying to characterize the set of first-order definable functions $f: \Bbb R \to \Bbb R$ and see what properties they have.
It is immediately clear that these functions are not all analytic ...
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What do you get if you perform the Dedekind Cuts procedure on $\mathbb Q(x)$?
Let $\mathbb Q(x)$ denote, as usual, the field of rational functions with rational coefficients. Any element of $\mathbb Q(x)$ can be written in the form
$$f=\frac{a_n x^n + \cdots +a_0}{b_m x^m + \...
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How can infinite sums be defined for a non-complete hyperreal field?
The definition of integration in non-standard analysis is $$\int_a^b f(x)dx:= st\left(\sum_a^b f(x)dx \right),$$ as given by Keisler (you could also use the transfer principle for internal functions). ...
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Are there guaranteed conglomerate sized models for a given first-order theory?
In a comment under What is the “maximal hyperreal field”? a commenter put “for any first-order theory 𝑇
with infinite models, one can prove the following in NBG set theory with the axiom of global ...
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How to use non-standard analysis to prove Baire Category Theorem？
I'm caring about some questions of non-standard analysis. I have found the only book talking about Baire Category Theorem, which is the book of Siu-Ah Ng. But I think the proof in this book is not ...
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Noetherian Rings in Nonstandard Framework
I have been trying to go through some algebraic geometry using the nonstandard framework. Noetherian rings are of course fundamental in this subject and it is characterized by the attribute that every ...
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Equivalence of the nonstandard analysis integral and the Riemann integral
I have a question about the definition of the integral in nonstandard analysis. The definition that I've usually seen is this: given a function $f(x)$ that you want to integrate from $a$ to $b$, you ...
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Is this a valid integral to prove area of circle?
Similar to the original poster of this question Is this a valid proof for the area of a circle?, I am a high school AP Calc BC student using the idea of Riemann sums to add an infinite number of ...
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Does non-standard analysis lead to different PDEs from those obtained through standard analysis?
Take the Navier-Stokes equations as an example. If we take a non-standard analysis approach, will the final form of the PDEs be different from what presented in classical books on fluid dynamics?
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Distribution that is 0 everywhere but sums up to unity?
For instance say I have random variable $X \sim \mathcal{N(\mu,\sigma)}$ Now the $\lim\limits_{\sigma \to \infty} f_{X} = 0$, so unfortunaltly it's not a pdf. However infinitesimally smaller values ...
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What is the “maximal hyperreal field”?
In many SE posts and the Wikipedia article on the surreal numbers I’ve seen references to a “maximal” hyperreal field that’s isomorphic to the surreals. If they’re isomorphic, then why is it that ...
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How can a field be Cauchy complete and non Archimedean
The Wikipedia page for the completeness of the Real numbers, says that “ there are non Archimedean fields that are ordered and Cauchy complete.” However, in many other places, I’ve read that non ...
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Cardinality of the set of internal sets
Non standard analysis relies on using sets that have the same properties as sets of real numbers to transfer many good theorems over. Every set $S\in P(\Bbb R)$ has a unique extension $S^* \in P(\Bbb ...
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Nonstandard algebraic geometry: Fundamental Theorem of Algebra
I have been trying to study the basics of algebraic geometry using nonstandard analysis and I can't wrap my head around this issue.
Let $^*\mathbb{C}$ be the extension of the complex numbers. Now ...
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Arithmetic in a nonstandard model of Real numbers
A problem from Section 3.3 in "A Friendly Introduction to Mathematical Logic" (Christopher C. Leary, Lars Kristiansen; 2nd edition):
(a) In the structure $\mathfrak{A}$ that was built in ...
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Repeated transfer principle for transfinite induction.
In set theory, one can use a non-principal ultra filter, $U_0$, to construct the hyperreal numbers. $\mathbb{R}^*$ is constructed as $\mathbb{R}^{\mathbb{N}}/U_0$. The transfer principal means that ...
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Is $No^2$ or $(\mathbb{R}^*)^2$ rotation invariant?
The fields of the surreal and hyperreal numbers aren’t complete. I’ve noticed that $\mathbb{Q}^2$ isn’t rotationally complete as $(0,1)$ could be rotated to a point not in the rational plane (like $(1,...
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Can the hyperreal numbers have a property akin to completeness by considering hypernatural sequences?
I’ve seen that $\mathbb{R}^*$ isn’t complete as many Cauchy sequences won’t converge, and that includes power series. In other stack exchange posts, I’ve seen that even the exponential function won’t ...
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Can the hyper hyper real numbers be constructed?
The hyperreal numbers can be constructed as $\mathbb{R}^{\mathbb{N}}/U$ given some ultra filter and this allows first-order statements to be transferred over to $\mathbb{R}^*$. Can this be done again ...
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Approach to construction hyperreal number with sequence
I read that hyperreal numbers can be constructed with sequences. For example, $\varepsilon = (1, 1/2, 1/3, ...)$ and $\varepsilon$ is infinitesimal.
But there is no smallest number (infinitesimally ...
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Number of models of the naturals and reals with and without CH
The Wikipedia page on true arithmetic says that it has $2^\kappa$ models for each uncountable cardinal $\kappa$. This refers to the theory of all first-order statements of the naturals.
I'm curious ...
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How is it possible that $dx$ contains $dx^2$ $N$ and $N+1$ times simultaneously?
Let's assume that $N$ is even infinite hyperinteger and $N=1/dx$, $N=dx/dx^2$.
Let's assume that we have expression $dx+dx^2$. We can add up $dx^2$ to $dx$ and $dx$ is obtained: $dx+dx^2=dx$ (...
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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 ...
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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 ...
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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 ...
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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 ...
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"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," ...
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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 ...
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How do we naturally extend the average from the Hausdorff Measure for functions with a domain without a gauge function?
Motivation: According to this question
Some sets have a Hausdorff Dimension $\alpha$ but have a
zero-dimensional Hausdorff Measure. These sets may have another
dimension function, i.e. a function $h:[...
user1027188
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Nonstandard proof of Lebesgue Number Lemma
I am currently going through Munkres's Topology but trying to prove things with nonstandard methods. The one I am struggling with right now is the Lebesgue Number Lemma, not too sure if my proof is ...
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How to derive the triple product rule with Nonstandard Analysis?
$\frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} \cdot \frac{\partial z}{\partial x} = -1$ according to the triple product rule
However, it would be 1, if derivatives behaved like ...
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Are all Hyperreal Infinitesimals representable by Monotonically Decreasing Sequences to 0?
I know there are many possible theoretical ways to built *R, including axiomatic and set-theoretic approaches. I am limiting my attention specifically to the Superstructure approach, perhaps best ...
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