https://lawrencecpaulson.github.io/2022/08/10/Nonstandard_Analysis.html

[utterances-bot]

The formalisation of nonstandard analysis

https://lawrencecpaulson.github.io/2022/08/10/Nonstandard_Analysis.html

[meithecatte]

I am confused as to how the hyperreals being a non-standard model of the reals fits in with the theorem that says the complete ordered field is unique up to isomorphism...

[lawrencecpaulson]

The point is the lack of expressivity of first-order logic. It cannot uniquely characterise any infinite structure, not even the natural numbers.

[CoghettoR]

Please note that "Elementary Calculus: An Infinitesimal Approach", by H. Jerome Keisler is freely available (licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License) since 2012.

"A Third Edition of this book was published by Dover Publications, Inc. in 2012, with the agreement that this online version will continue to be freely available. This gives you the choice of downloading this free version or purchasing the printed book." Source: https://people.math.wisc.edu/~keisler/calc.html

[sterraf]

@meithecatte They are model of the first-order properties of the reals. That includes being an ordered field and more, but completeness is not first-order: it states “for all subsets...” And the existence of hyperreals is a proof the fact that it is not first-order! The same goes for Archimedeanity.