I Several Questions About Smooth Infinitesimal Analysis

[Mike_bb]

Hello.
I read about smooth infinitesimal analysis and I have several questions:

1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6)

2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2) (https://www.fuw.edu.pl/~kostecki/sdg.pdf , page 21)

Thanks!

 

[pbuk]

I read about smooth infinitesimal analysis and I have several questions:

It is not possible to understand SIA without solid understanding of logic, set theory and real analysis; do you have these?

1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6)

This is standard notation (although perhaps not typeset very well). $\varepsilon \cdot 1$ means ε multiplied by 1. Photo 1 is part of a proof that the law of the excluded middle (LEM) does not hold in SIA.

2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2) (https://www.fuw.edu.pl/~kostecki/sdg.pdf , page 21)

I don't understand what you are asking. In photo 2 the author uses the Kock-Lawvere Axiom and LEM to derive a statement that is a contradiction, again proving that LEM does not hold in SIA.

What is your interest in SIA? Are you really comfortable working in a domain where $x^2 = 0$ does not mean $x = 0$? Or where $x \ne 0$ being not true does not mean $x = 0$?