Questions tagged [nonstandard-analysis]
Nonstandard analysis is a way of doing calculus and analysis with infinitesimals. The historical approach of Leibniz, Euler, and others to infinitesimal calculus was gradually replaced by epsilon, delta techniques in the context of a real continuum, in the 19th century. It was not until the 1960s that Abraham Robinson developed a theory of a hyperreal continuum that allows for a development of analysis procedurally akin to that of its founders.
128 questions
5
votes
3
answers
901
views
Non-standard analysis and higher-order model theory
$\newcommand\SThing[1]{\Sigma\text-\mathrm{#1}}\newcommand\SType{\SThing{Type}}\newcommand\SFun{\SThing{Fun}}\newcommand\SRel{\SThing{Rel}}$I am about to begin my doctorate in philosophy, and my ...
- 181
8
votes
2
answers
993
views
Is an ultrapower essential for defining the hyperreals?
If AC holds, the hyperreals are typically defined using the ultraproduct construction.
Without AC, such as in ...
- 1,465
3
votes
0
answers
165
views
Meta-principle for approximation of kernels
Suppose, we have an approximation result for finite matrices of all orders $n \times n$. When can we push such a result in the case of infinite matrices or kernels, maybe via possibly some ultralimit ...
- 61
7
votes
1
answer
449
views
Proper class sized hyperreals
In his paper [1], pp. 36-37, Ehrlich quotes Keisler on a possible construction of proper class sized hyperreals. Keisler indicates to construct four objects $\mathbb{R}$, $\mathbb{R}^*$, $<^*$, $^*$...
- 73
-1
votes
1
answer
202
views
Homomorphism from field of hyperreals to field of reals? [closed]
I am curious if it is possible to construct a homomorphism from a field of hyperreal numbers to the field of real numbers? (Similarly, a homomorphism from the surreals to the reals?)
Assuming that ...
- 65
17
votes
2
answers
2k
views
Do the surreal numbers enjoy the transfer principle in ZFC?
The surreal field $\newcommand\No{№}\No$ is definable in ZFC, and it is easy to see that the surreal order is $\kappa$-saturated for every cardinal $\kappa$, precisely because we fill any specified ...
- 247k
5
votes
0
answers
117
views
Definable pseudo-standard predicates in Internal Set Theory
Consider the usual language $\mathcal{L}=(\in, \mathrm{st})$ of Nelson's Internal Set Theory, and a unary $\mathcal{L}$-predicate $P$. For an $\mathcal{L}$-formula $\varphi$, let $\varphi^P$ denote ...
- 783
4
votes
3
answers
777
views
In hyperreal field, can ln(ε) and ln(ω) be expressed as infinite sums?
In the hyperreal field, we can use Taylor series to express e^(ε) and e^(ω) as:
e^(ε) = 1 + ε + (ε^2)/2! + ...
e^(ω) = 1 + ω + (ω^2)/2! + ...
Is it similarly possible to express ln(ε) and ln(ω) as ...
- 65
8
votes
1
answer
373
views
Finitistic interpretation of Nelson's internal set theory
What does “standard” in internal set theory really mean?
Is it secretly a way of reconciling conventional mathematics with (ultra)finitism?
Until recently I thought “standard” was just a way of ...
- 17.6k
12
votes
2
answers
2k
views
Am I doing a forcing argument here?
I have an argument of the following form:
Executive Summary:
We have a $\mathbb R$-valued function $L$ which we want to show is $\mathbb Z$-valued. We approximate it by $\mathbb Q$-valued functions $\...
- 66.8k
4
votes
1
answer
239
views
Automorphism on the hyperreals
$\DeclareMathOperator\hal{hal}$A field isomorphism $\phi:F\rightarrow G$ is a bijection such that (i) $\phi(x+y)=\phi(x)+\phi(y)$ and (ii) $\phi(xy)=\phi(x)\phi(y)$, where $F$ and $G$ are ordered ...
- 151
5
votes
1
answer
337
views
What is the theory of computably saturated models of ZFC with an *externally well-founded* predicate?
For any model of $M$ of ZFC, we can extend it to a model $M_{ew}$ with an "externally well-founded" predicate $ew$. For $x \in M$, We say that $M_{ew} \vDash ew(x)$ when there is no infinite ...
- 7,158
4
votes
1
answer
617
views
Paris-Harrington via overspill?
I saw in an old logic paper that the Paris-Harrington theorem can be proved via Overspill. The presentation is unfortunately too technical for me to follow. Does somebody have any insight into this?
- 18.1k
20
votes
1
answer
2k
views
Does every series of hyperreal numbers converge to some hyperreal number?
I am currently trying to find some field $F$ which includes $\mathbb{R}$ (or $\mathbb{C}$) and in which series $x^* = \sum_{i\in\mathbb{N}} x_i$ converge to some element of the field. (i.e. $x^* \in ...
5
votes
1
answer
330
views
Are there results unique to non-standard analysis or surreal numbers that have not been reconciled with classical analysis?
I am exploring areas where non-standard analysis or the theory of surreal numbers has yielded results that remain exclusive to these frameworks without analogs or proofs in classical analysis. For ...
7
votes
0
answers
159
views
Points of the sheaf topos over Blass' category
There is a site $\textsf{Blass}$ used for (constructive) non-standard analysis, whose objects are sets equipped with a filter, and morphisms are continuous functions defined up to a small set. (It is ...
- 2,166
3
votes
0
answers
227
views
In constructive set theory, is it consistent for there to be a ring that models smooth infinitesimal analysis?
In a constructive set theory such as CZF, it is consistent to assume that every function $f : \mathbb R \to \mathbb R$ is continuous. However, it is not consistent to assume that every such function ...
- 7,158
7
votes
5
answers
2k
views
Standard models of N and R: An Alice/Bob approach
This is a question about a comment in a recent publication by Roman
Kossak. Kossak wrote:
"Nonstandardness in set theory has a different nature. In
arithmetic, there is one intended object of ...
- 18.1k
2
votes
0
answers
403
views
Comparison of model-theoretic and axiomatic approaches to NSA
This question is motivated by the discussion in the comments to this
post. The question concerns
a comparison of model-theoretic (extension) approaches to nonstandard
analysis, and axiomatic (...
- 18.1k
6
votes
1
answer
476
views
Quantifier complexity of definition of compactness
This question is inspired by the post on quantifier complexity of
continuity. We work with metric spaces M
considered as two-sorted first-order structures (M,$\mathbb R$,d,+,⋅,<)
where $d:M^2→\...
- 18.1k
13
votes
1
answer
836
views
Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC?
We work in ZFC.
Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not field-isomorphic to $\mathbb{R}$. For example,...
- 2,389
4
votes
1
answer
249
views
Realization of $\mathbb{R}((X))$ as a subquotient of a hyperreal field ${}^{*}\mathbb{R}$
Now we fix an ultrafilter of $\mathbb{N}$ that contains the cofinite filter, consider a hyperreal field ${}^{*}\mathbb{R}$. Let $\varepsilon$ be a positive infinitesimal. We doubt that a power series ...
- 489
10
votes
3
answers
654
views
Is there a constructive version of internal set theory?
Is there a theory T such that:
T includes all the axioms of CZF.
T includes the Idealization, Standardization, and Transfer schemas from IST.
Every axiom of T is a theorem of IST.
T has Church's rule....
- 7,158
9
votes
1
answer
384
views
Transfinitely iterating the Levi-Civita, Hahn or Puiseux constructions
This question was originally asked at MSE but seems too advanced, so I'm reposting it here.
In short, the idea is that many constructions for non-Archimedean fields can naturally be iterated, in some ...
- 5,005
16
votes
3
answers
2k
views
Interpreting Conway's remark about using the surreals for non-standard analysis
In Conway's "On Numbers And Games," page 44, he writes:
NON-STANDARD ANALYSIS
We can of course use the Field of all numbers, or rather various small
subfields of it, as a vehicle for the ...
- 5,005
0
votes
1
answer
168
views
Levi-Civita field in unusual basis
Can all elements of the Levi-Civita field be represented as power series of a single element
$$p=\varepsilon^{-1}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}...
- 1
1
vote
2
answers
266
views
In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?
In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?
Particularly, is there an element $w$ of the field such that the ...
- 1
0
votes
1
answer
241
views
Automating proofs via indicator functions?
The following is a cross-post of this question on math.SE, which did not attract any comment and may therefore be too research-oriented for math.SE.
It is a common technique in measure theory to ...
- 293
4
votes
1
answer
388
views
On a completeness property of hyperreals
Let $\mathbb{R}_*=\mathbb{R}^\omega/\mathcal U$ for some ultrafilter $\cal U$. In the definitions of this question and assuming ZFC + CH there are only three types of cuts in $\mathbb{R}_*$: $(\omega,\...
- 1,153
10
votes
2
answers
1k
views
SPOT as a conservative extension of Zermelo–Fraenkel
In Infinitesimal analysis without the Axiom of Choice, Hrbacek and Katz have shown that it is possible to formulate an axiomatic theory which provides a formalisation of calculus procedures which make ...
- 5,481
3
votes
0
answers
205
views
Is anything known about $\Delta_n$ bounding?
For a class $\Gamma \in \{ \Sigma_n, \Pi_n, \Delta_n \}$ in the arithmetical hierarchy, we can consider the induction, bounding, and least number principles for $\Gamma$:
$\mathsf{I}\Gamma$ is $\big[ ...
- 828
2
votes
1
answer
258
views
Can $\mathsf{RCA}_0$ prove that every nonempty c.e. set $A \subseteq \mathbb{N}$ has a least element?
In other words, can $\mathsf{RCA}_0$ prove that for every function $f\colon \mathbb{N} \to \mathbb{N}$, there is $b \in \mathbb{N}$ such that
$$ \exists k \in \mathbb{N},\ f(k) = b\quad \land\quad \...
- 828
1
vote
0
answers
77
views
Legitimacy of the shadow map serving as a field homomorphism for a specific hyperfinite field formed of a union of hyperfine lattices
I'm hoping to get some comment on the legitimacy of my approach to creating a hyperfinite ring formed of a union of modular groups in order to obtain a field homomorphism from this hyperfinite space ...
- 141
1
vote
1
answer
188
views
What's the size of non standard monad for weak topology?
There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space):
$$\mu(0) = \{y\in{}^{*}X: \forall x\in X ~~ \...
- 75
2
votes
0
answers
133
views
Request for bibliographic information
Greetings to everyone on this forum (I am a new-comer). I would like to ask the experienced members for suggestions on (as) comprehensive and systematic (as possible) bibliographic sources regarding:
...
- 121
4
votes
1
answer
415
views
Loeb measures and non-standard hull of Banach spaces
$\DeclareMathOperator\Fin{Fin}$I am trying to understand the notion of "liftings" of Loeb measurable functions to internal, internally measurable functions, and its connection to non-...
- 959
12
votes
2
answers
680
views
Decidability of a first-order theory of hyperreals
The theory of real closed fields is decidable. The hyperreals satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals.
If we add a unary ...
- 7,158
2
votes
1
answer
505
views
Unbounded $\omega_1$-sequence in $^*\mathbb{N}$
Let $\mathcal{F}$ be a non-principal ultrafilter on $\omega$. Let $^*\mathbb{N}$ = $\mathbb{N}^\omega/\mathcal{F}$ be an ultrapower. Let $\{n_\alpha\}_{\alpha\in\omega_1}$ be a strictly increasing ...
6
votes
0
answers
225
views
Isomorphism of hyperreal fields viewed as extensions of the field of reals
I asked this question on Mathematics Stackexchange but got no answer.
Question. Does $ZFC$ prove that there are non-principal ultrafilters $\mathcal U$ and $\mathcal V$ over $\mathbb N$ such that the ...
- 4,456
9
votes
2
answers
491
views
Can nonstandard fields contain $\mathbb R$ in different ways?
Suppose $e : \mathbb R \to F$ is an elementary embedding in the language of ordered fields. Can there exist an elementary embedding $e' : \mathbb R \to F$ such that $e \not= e'$? Note that it would ...
- 21k
6
votes
1
answer
414
views
Cofinality of infinitesimals
Suppose $\kappa$ is an infinite cardinal and $U$ is a countably incomplete uniform ultrafilter over $\kappa$. Then $\mathbb R^\kappa/U$ is nonstandard. What is the cofinality of the set of ...
- 21k
6
votes
0
answers
241
views
Influence of cardinal characteristics on nonstandard analysis?
As I understand, nonstandard analysis usually proceeds by taking a ultrapower of the universe by some nonprincipal ultrafilter on $\mathbb N$. There are continuum many “integers” of this model, but ...
- 21k
2
votes
0
answers
297
views
Continuum hypothesis in nonstandard universe
In Vladimir Kanovei's book "Nonstandard Analysis, Axiomatically", some nonstandard set theory is introduced. It seems that, one of them, DNST, is useful.
When we are talking about higher order ...
- 475
3
votes
1
answer
307
views
Ultrapower of amenable group
Let $\Gamma$ be an amenable group. Consider its ultrapower $^*\Gamma$. It is known that $^*\Gamma$ need not be amenable. In fact, there is a stronger notion of uniform amenability for $\Gamma$ (...
- 959
2
votes
0
answers
193
views
How to construct "inaccessible hypernatural"?
Consider that, take a sufficient large natural number $a_1$, then take a natural number $a_2$ sufficient large to $a_1$, then take $a_3$,...
Now we have a function $n \mapsto a_n$ which grows very ...
- 475
2
votes
0
answers
187
views
Embedding standard function spaces into superstructure
I have a question concerning the precise handling the usual function spaces like $L^2$ in the context of the superstructure. In their paper
Benci, Vieri; Luperi Baglini, Lorenzo. Generalized ...
- 18.1k
4
votes
0
answers
190
views
Self homomorphisms of hyperreals fixing the reals
What do we know about the circumstances (whether having to do with the axioms of set theory or the model itself) under which a field $F$ of hyperreals (=ultrapower of $\mathbb R$ with respect to a non-...
- 3,077
9
votes
2
answers
1k
views
A definition of topology using monads (a.k.a. halos)
In nonstandard analysis, there is a way of studying topological spaces known as "monads" (more commonly known as halos, as it turns out). The monad of a point $x$ (written $\mu(x))$ is the set of all ...
- 7,158
24
votes
1
answer
2k
views
Has anything (other than what is in the obituary written by M. Noether) survived of Paul Gordan's defense of infinitesimals?
Question. Has anything other than what can be guessed from this obituary written by Max Noether survived of the 'defense' of infinitesimals that Paul Gordan gave in his doctoral disputation on March 1,...
- 6,151
8
votes
1
answer
647
views
What is the Turing degree associated with an ultrafilter $U$?
I asked Turing degree of a turing machine with access to an (arbitrary) nonstandard integer, not thinking about the possiblity that this could depend on the model used. The question was not formulated ...
- 7,158