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Results tagged with nonstandard-analysis
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user 45005
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.
7
votes
Accepted
On a completeness property of hyperreals
This is also called Cauchy-completeness, and it coincides for non-Archimedean ordered fields with the natural valuation to the valuation-theoretic notion of completeness. Also, this is the same as hav …
- 2,519
2
votes
Can nonstandard fields contain $\mathbb R$ in different ways?
In fact $\mathbb{R}$ is elementarily embedded in several ways in any non-archimedean real-closed field which contains it. The proof is more involved than I thought before writing it, but if you don't …
- 2,519
4
votes
Accepted
Unbounded $\omega_1$-sequence in $^*\mathbb{N}$
If HC (continuum hypothesis in French) holds, then some of those sequences are cofinal whereas some are not.
Indeed, HC implies that the corresponding ultrapower$\ ^*\mathbb{R}$ of $\mathbb{R}$ is a s …
- 2,519
7
votes
Realization of $\mathbb{R}((X))$ as a subquotient of a hyperreal field ${}^{*}\mathbb{R}$
Let $A$ denote the convex subring of hyperreal numbers $y$ for which there exists an $n \in \mathbb{N}$ with $-\varepsilon^{-n}<y<\varepsilon^{-n}$. This has the set $\mathfrak{m}$ of numbers $z$ with …
- 2,519
8
votes
Accepted
Transfinitely iterating the Levi-Civita, Hahn or Puiseux constructions
Let us work in NBG set theory with global choice. There is, up to non unique isomorphism, a unique real-closed field that is $\kappa$-saturated for all infinite cardinals $\kappa$. Let's denote it by …
- 2,519