[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.
Tim
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