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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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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 …
Ali Enayat's user avatar
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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 $PA$ …
Ali Enayat's user avatar
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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 …
Ali Enayat's user avatar
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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 …
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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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