[FOM] Wildberger on Foundations

matthias matthias.eberl at mail.de
Mon Dec 3 16:19:58 EST 2018

Timothy Chow wrote

Question 1. To what extent can we develop modern analysis on the basis 
of PA alone (or perhaps even Heyting arithmetic), without any reference 
to infinite sets? Can this be done in an undergraduate-friendly way 
without a lot of extra baggage from logic?

A side note to your question: It seems to me that completed infinite 
sets are mainly a matter of semantics, not so much of the theory. So PA 
requires an infinite set (the actual infinite set of all natural 
numbers) if interpreted within the common Tarskian semantics. On the 
other hand, you can do ZFC set theory (as a first order theory) in a 
purely finitistic way --- see the section "The Finite Mathematics of 
Indefinitely Large Size" in Shaughan Lavine's book "Understanding the 
Infinite" (this approach is however not "undergraduate-friendly").

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