[FOM] Wildberger on Foundations

Timothy Y. Chow tchow at math.princeton.edu
Mon Dec 3 20:53:47 EST 2018

matthias wrote:

> 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.

It may have been a mistake for me to mention formal systems in my 
statement of Question 1, because it may have created the wrong impression.

When I was an undergraduate, the "rigorous" introduction to analysis 
defined the real numbers as a complete ordered field with the least upper 
bound property, and constructed it using Dedekind cuts (or maybe Cauchy 
sequences---I don't remember).  Dedekind cuts are of course infinite sets.

The question is whether we can dispense entirely with all the traditional 
constructions of real numbers, and still write a textbook for a "rigorous 
first course in analysis."  Formal systems such as PA or EFA or whatever 
should not intrude into the foreground---current undergraduate texts on 
analysis don't mention formal systems, after all.

I don't think it's possible to talk about analysis without functions, so 
functions have to be introduced somehow, presumably as "rules" rather than 
as infinite sets of ordered pairs.  One of the main challenges is to 
figure out how to avoid talking about the set of real numbers or even an 
"arbitrary real number," without introducing unnatural circumlocutions.


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