Hacker News Reader

Comment by v64 | Hacker News Reader

If you reject the axiom of choice, what are your thoughts on Zorn's lemma and the well ordering theorem (other than the observation of "they too are false" via equivalency)?

View on HN

The ultrafilter attempt to bypass Arrow's Impossibility Theorem demonstrates how Zorn sends you up the creek without a paddle.

You can postulate such an object exists but you cannot realize it, so it doesn't translate to praxis. (e.g. you can't use the ultrafilter to decide an election)

That which can be constructed or described in a finite number of bits is more real than the phony numbers that Cantor justified. (e.g. Feigenbaum's constant is more real than any one of those real numbers that classical analysts try to bracket but never catch)

I got my honorable discharge from grad school and part of the climb in mathematical physics is reading some paper from 1957 that looked promising but after a close read you learn they got it wrong at page 47 and you have to figure it out yourself because you can't find the answers in the literature. You find out that the median scientific paper is wrong the hard way.

Wolfram wants to use computation (e.g. simulation, construction) as a praxis for all intellectual activity so he should privilege that map out of the Borges story over the territory of that deteriorating Empire which it mirrors.

Scientists in 2020 don't calculate in Cantor's phony numbers, but instead with those IEEE floats which never work quite right when you decimalize them.

> Feigenbaum's constant is more real than any one of those real numbers that classical analysts try to bracket but never catch

What do you mean?

Feigenbaum's constant is a number like Pi or e. It is transcendental. Even though you can't write it down with a finite number of digits, you can write down a formula to compute as many digits as you want (if you are ready to boil the oceans, build a Dyson sphere, harness a Quasar) You can give it a name and refer to it directly.

Any formula like that provides a set of brackets, "real" numbers with a finite number of digits (e.g. names) that we can say that the "phony" number is between. We can make the brackets finer and finer, but you can't pick out one in particular.

Thus 3, pi, pi/e + 6, sqrt(pi-e) are more "real" than the the continuum we imagine between them. Being able to name things, for instance, makes it possible to talk about them.

Constructivist mathematics [1] is an approach to math that restricts the universe of discourse to objects that can be explicitly defined. With this restriction, the subset of the real numbers considered consists of the reals that are definable [2], such as those that are computable [3] and constructible [4].

Similarly, the axiom of choice allows for the existence of nondefinable choice functions [5] in certain cases, so is rejected.

Regarding the part about analysis, the field of computable analysis [6] exists to establish analysis on constructivist footing.

[1] https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...

[2] https://en.wikipedia.org/wiki/Definable_real_number

[3] https://en.wikipedia.org/wiki/Computable_number

[4] https://en.wikipedia.org/wiki/Constructible_number

[5] https://en.wikipedia.org/wiki/Choice_function

[6] https://en.wikipedia.org/wiki/Computable_analysis

Better: what do you think of "the product of nonempty sets is nonempty".

I believe this statement only requires the axiom of choice if the number of sets is infinite.

Reply on news.ycombinator.com