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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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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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 ...
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How is compactness related to countable saturation?
By Cantor's intersection theorem every decreasing nested sequence of nonempty compact sets has a common point.
A superficially similar result holds that every decreasing nested sequence of nonempty ...
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Archimedean completeness of some fields
I need a reference (different from Hahn's 1907 paper) for the following result.
Theorem: If $G$ is a totally ordered abelian group, then the field $\mathbb{R}((G))$ is archimedean complete.
$\...
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Compactness and omega models
If $T$ is a first order set theory having finitely many axioms, suppose the consistency of $T$ is already known and that $T$ proves existence of naturals, now suppose that $S$ is a schema and that $T+...
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What are the minimal requirements for the definable hyperreal field plus transfer?
It is interesting that to prove the transfer principle for the definable hyperreal field, one requires no more choice than for proving, for instance, the countable additivity of the Lebesgue measure. ...
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Is there a model of ZF+ACC where transfer fails for the definable hyperreals?
In 2003 Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go ...
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Transfer with minimal choice
Let FUF postulate the existence of a Free UltraFilter on $\mathbb{N}$ and ACC the axiom of countable choice. Consider the superstructure on $\mathbb{R}$ and its inclusion in the bounded ultrapower. ...
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Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?
The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form.
Is there possible an extension of real/complex numbers in which logarithms and ...
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What's Reeb's take on naive integers?
Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models ...
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Who said "the naive counting numbers don't exhaust $\Bbb N$"?
In the context of Robinson's framework, or more precisely its reformulation by Ed Nelson, one of the practitioners in the field expressed the sentiment something like "the naive counting numbers don't ...
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What is the modern consensus on the difficulty of infinitesimals?
At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete ...
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What is... a grossone?
Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The ...
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