Defining "Penon Infinitesimals".

Asked Mar 15 '14 at 17:54

Active Mar 25 '14 at 15:52

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In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon infinitesimals", the set of which (I interpret as being) "defined" as follows:

$\{x\in\mathbb{R}\mid \neg\neg(x=0)\}$.

These look phenomenal. I'd like to (confirm that this is indeed the definition and) find out more about them but nearly everything I've seen on them is behind some pay-wall or another. So, what exactly are they?

edited Mar 25 '14 at 15:52

asked Mar 15 '14 at 17:54

Shaun

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Search for Smooth Infinitesimal Analysis, and the book A Primer of Infinitesimal Analysis by Bell. – Matthew Towers Mar 15 '14 at 17:59
Beautiful, thank you! $\ddot\smile$ – Shaun Mar 15 '14 at 18:04
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@mt_ "Penon" is not mentioned anywhere in the suggested reference. – Andrés E. Caicedo Mar 15 '14 at 18:16
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But I can confirm that your interpretation is correct. See here. (This is from "Mathematical logic and theoretical computer science", D. Kueker et al., eds., pg. 119. The original reference is this one.) – Andrés E. Caicedo Mar 15 '14 at 18:21
Thank you! That's perfect (and the comment by @mt_ was really helpful nonetheless). – Shaun Mar 15 '14 at 18:29
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Also check out a relevant fine blog post by Andrej Bauer (if you are in a hurry, scroll down to Smooth infinitesimal analysis). If you want more details, check out Mike Shulman's pizza seminar notes. They are very accessible. – Ingo Blechschmidt Mar 16 '14 at 11:32
Thank you, @IngoBlechschmidt. That's all fascinating. I've removed the "nonstandard-analysis" tag now thanks to the blog post :) – Shaun Mar 19 '14 at 13:42

Defining "Penon Infinitesimals".

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