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).

374 questions
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Does nonstandard analysis allow for a more powerful second derivative test?

The second-derivative test states that if $x$ is a real number such that $f'(x)=0$, then: If $f''(x)>0$, then $f$ has a local minimum at $x$. If $f''(x)<0$, then $f$ has a local maximum at $x$...
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Is the union of non-standard analogs of a family of sets a proper subset of the non-standard analog of the union of those sets?

The book on formal logic I'm using for self-study builds the foundation for non-standard analysis (to illustrate the usage of non-standard models) using the following construction. Consider the real ...
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Is this proof correct: A nonstandard-statement transfers exactly iff it's an internal set.

Let $\varphi\in \widehat {^*S}$ be a statement in a nonstandard-superstructure. The following proof is supposed to show that if and only if $\varphi$ is internal, there exists a corresponding ...
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Non-standard integers?

Edit: Realized almost immediately that this was a stupid question - see comment below. Was about to delete it when an answer appeared that seems like saving... Context: I just saw a presentation of ...
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What's wrong with the following (Robinson-) nonstandard proof?

Note: I do not know for sure that it's wrong, but have a strong suspicion, as the authors implicitly mentioned it but in the end chose another (more elaborate) proof. Let $\hat S$ be a superstructure....
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How do I prove that every hyperreal has a standard part after constructing the reals from the hyperrationals?

In texts on nonstandard analysis, I've come across references to the following construction of the real numbers: starting from the hyperrationals $^*\mathbb Q$, say that $\mathbb R$ is the quotient ...
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Completeness in Levi-Civita field

I've been wondering for quite a time about Levi-Civita field (you can read it simply in https://en.wikipedia.org/wiki/Levi-Civita_field). I remember that I've read somewhere that Levi-Civita field is ...
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Is the Archimedean principle necessary to prove the density of $\mathbb Q$ in $\mathbb R$?

I've noticed that most proofs of the density of $\mathbb Q$ in $\mathbb R$ use the Archimedean principle. For example see @Arturo Magidin's checked answer here. Density of irrationals I'm puzzled by ...
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Can we ever reach $\mathbb{R}$ from $\mathbb{R}*$? [closed]

The construction of the hyperreals is quite the ways over my head, but, after watching a series of videos that try to conceptualise the hyperreal number line from sources such as mainly YouTube and ...
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Which branch of mathematics rigorously defines infinitesimals?

I have some trouble doing standard computations in calculus because of the notion of a differential, otherwise known as an infinitesimal, being rather ill defined, in my experience. Are there any ...
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If $f(q)=g(q)$ for all rationals, prove that $f=g$ by nonstandard methods.

I am trying to prove that if $f,g:\mathbb R\to\mathbb R$ are ordered ring homomorphisms, then, if $\forall q\in\mathbb Q,f(q)=g(q)$, then $f=g$. Is it true ? Can you give a nonstandard proof of this ...
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Unitary operator as an infinitesimal transformation

Somebody can give me a hand with this? Given the unitary operator $U=1+i\varepsilon F$ (where $\varepsilon$ is an infinitesimal escalar), in order to prove that $F$ is Hermitean: $$UU^{+}=1$$ (1+...
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In hyperreal, does EVP imply IVP? Other way?

So I define those two properties as ($\mathbb{R_H}$ denotes hyperreal numbers): EVP: If $I$ is an interval and $f:I\rightarrow\mathbb{R_H}$, we say that $f$ has the extreme value property iff $f$ has ...
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What is the relation between $\varepsilon$-$\delta$ and $dy$-$dx$ notations?

For what I know, $dy$-$dx$ aren't real numbers, exist as convenient notations to capture our intuitions about infinitesimal increments, while $\varepsilon$-$\delta$ are real distances, saying that $f$ ...
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Does the transfer principle really work in both directions?

Let $\mathscr{S}$ be a statement in a superstructure $\hat{S}$. Let $^*$ denote the transfer of an element of $\hat{S}$ via the transfer principle. The transfer principle says that $\mathscr{S}$ is ...
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Can any Neutrix be written as a countable intersection or union of internal sets? (IST)

I need help understanding a counter-example in non-standard analysis. First a few preliminaries: Def.: A Neutrix is a convex additive subgroup of the hyperreals ${}^*\mathbb R$. The two simplest ...
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How can a universe include an infinite (sum) relation

Let the universe be $\hat{V}$, which is constructed as: $\hat{V} := V_0 \cup V_1 \cup V_2\cup ...$ where $V_0$ is the set of primary elements and $V_{v+1} := V_v \cup P(V_v)$ So, in other words,...
Consider the following funny argument: Let $d$ be the well-known exterior derivative. We know for any $k$-Form $f \in \Omega^k(M)$ we have $ddf=0$ and $d$ is a linear map. Then $df$ must already have ...