Several Questions About Smooth Infinitesimal Analysis

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Mike_bb
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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!
 

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pbuk
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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 ##?
 

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