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Results tagged with nonstandard-analysis
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user 12705
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.
6
votes
Accepted
What is the theory of computably saturated models of ZFC with an *externally well-founded* p...
Observe that if $M\models\def\zfc{\mathrm{ZFC}}\zfc$ has nonstandard $\omega$, then $x\in M$ is externally well founded iff its rank $\rho(x)$ is a standard natural number: this follows easily by iter …
- 47.1k
14
votes
Accepted
Decidability of a first-order theory of hyperreals
Yes, the theory is decidable.
If $F$ is an ordered field and $R\subseteq F$ a non-cofinal subfield, then
$$O=\{x\in F:\exists u\in R\:(-u\le x\le u)\}$$
is a convex valuation ring of $F$, with maximal …
- 47.1k
13
votes
Accepted
Can nonstandard fields contain $\mathbb R$ in different ways?
Yes, this is possible. Let $F$ be a nonarchimedean strongly $\omega$-homogeneous real-closed field such that $\mathbb R\subseteq F$ (which exists by model-theoretic general reasons). Fix an infinitesi …
- 47.1k
2
votes
Is a model of arithmetic contained in a model of arithmetic an initial segment?
Andres Caicedo has answered the question as stated, however let me point out that one can recover a form of the property under further assumptions on the pair of models. Namely, let $\mathbb N_1$ be a …
- 47.1k
2
votes
dense orders are saturated
This characterization for dense linear orders can be found as Proposition 16.1 in Sachs’ Saturated model theory. Note that he defines $\kappa$-density differently than you (this property is also known …
- 47.1k