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Results tagged with nonstandard-analysis
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user 9269
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.
10
votes
Paris-Harrington via overspill?
This edit includes a fleshed-out version of Smorynski's remark about taking advantage of a nonstandard model of arithmetic (instead of König's lemma) to prove the Paris-Harrington (PH) principle from …
- 17.7k
18
votes
Accepted
SPOT as a conservative extension of Zermelo–Fraenkel
In plain terms, the conservativity of SPOT over ZF means that if a particular statement S in the language of ZF is provable in SPOT, then ZF can already prove S (with a possibly different proof). Note …
- 17.7k
11
votes
Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups
This note complement Joel's, who pointed out that every first order theory with an infinite model has a model with many automorphisms (this was first proved by Ehrenfeucht and Mostowski in their famou …
- 17.7k
12
votes
What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a mode...
The theorem below answers question 4, and the corollary answers question 3. See also Theorem 2.
Theorem 1. For every pair of infinite cardinals $(\kappa$,$\lambda)$, there is a model $M$ of $PA$ …
- 17.7k
7
votes
dense orders are saturated
Philip Ehrich and Emil Jeřábek have given useful references (and Dave Marker and I seem to have simultaneously written our responses).
This note is to point out that the result itself was original …
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