# 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.

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### Actual infinitesimals for solving Vitali paradox

Has anyone tried to use actual infinitesimals to solve paradoxes about non-measurability? In Vitali paradox, for example, they divide a set with measure 1 into $\infty$ subsets of zero measure and ...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 (...

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### 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→\...

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### Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?

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 isomorphic to $\mathbb{R}$.
A field $E$ is ...

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### 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 ...

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### 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....

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### 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 ...

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### 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 ...

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### 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}...

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### 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 ...

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### 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 ...

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### 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,\...

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### 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 ...

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### 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[ ...

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### 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 \...

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### 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 ...

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### 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 ~~ \...

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### 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:
...

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### 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-...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\Gamma$ (...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 ...

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### 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,...

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### 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 ...

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### What is $\mu$-approximablity in Loeb measure (conflicting statements in books)?

In Loeb measure, a set is Loeb measurable iff it is $\mu$-approximable, where $\mu$ (roughly speaking) is a finitely additive hypervalued measure over internal sets.
But I found the definitions of $\...

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### Turing degree of a turing machine with access to an (arbitrary) nonstandard integer

Let us consider Turing machines (or other Turing-complete model of computation) that, in addition to their regular input, are given some integer $H$, where $H$ is positive nonstandard. This means, in ...

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### Did Lagrange change his mind about infinitesimals?

Lagrange is famous for his attempt to found analysis algebraically using power series expansions, an approach that, as we know today, is limited to analytic functions. Lagrange is also known as the ...

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### Was Cauchy prescient?

Cauchy proved a sum theorem for series of continuous functions in 1821, and published another article on the subject in 1853.
Michael Segre, writing in Archive for History of Exact Sciences, claimed ...

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### Non standard extension of real numbers via nonprincipal ultra filters

Assume That $U,V$ are two filters on the natural number $\mathbb{N}$.
We say that $U$ is equivalent to $V$ if there is a bijection $\phi: \mathbb{N} \to \mathbb{N}$ such that $\tilde{\phi}(U)=...

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### Did Bishop make those comments in his oral presentation?

The 1975 published version of a 1974 talk at a workshop by Errett Bishop contains the following comment:
"A more recent attempt at mathematics by formal finesse is non-standard analysis. I gather ...

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### Are the definable hyper-reals, using quantifiers only over the standard reals and natural numbers, the same as the algebraic numbers?

This question arose today at Yevgeny Gordon's talk, "Will nonstandard analysis be
the analysis of the future?" at the CUNY Logic
Workshop. Here is my way of asking it.
Consider the ordered real field ...

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### Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups

A classic result says the automorphism group of $\mathbb{R}$ (over $\mathbb{Q}$) is trivial. The proof is simple: every automorphism preserves squares, and hence fixes the positive reals, so it must ...