[FOM] Wildberger on Foundations
Rob Arthan
rda at lemma-one.com
Thu Dec 6 15:18:18 EST 2018
Tim,
> On 4 Dec 2018, at 01:53, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
>
> ...
> 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.
There is a paper that I am very fond of by Behrend advocating introducing real numbers via decimal expansions (which I guess can be introduced as "rules" for determining the n-th digit). Here's a BiBTeX reference for it:
@article{behrend56,
author="F.A. Behrend",
title="{A Contribution to the Theory of Magnitudes and the Foundations of Analysis}",
journal="Mathematische Zeitschrift",
volume=63,
pages="345-362",
year=1956}
Behrend's definition of addition involves quantifying over all the digits to the right of a given digit in order to describe how carries propagate when you have to work from left to right. He introduces multiplication by reasoning about order-preserving homomorphisms of the additive group: but homomorphism are just functions, so it could well be possible to fit his approach in with your line of thinking.
Regards,
Rob.
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