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0 pointsMichaelBurge9y ago0 comments

And then there's the Axiom of Choice and all that it brings. There's a Hamel Basis for the reals which is provably impossible to construct(since it's equivalent to AC).

Do you believe in the existence of things that are provably impossible to construct?

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btilly9y ago

As it happens, I don't.

That said I'm quite able to quote and do mathematics that depends on axioms that I do not accept. I can be quite the formalist when it is convenient. But I privately think of it as formal bullshit. And don't like seeing people who haven't accepted it beaten over the head until they do.

Smaug1239y ago

For that matter, do you believe in really, really big integers?

MichaelBurgeOP9y ago

There are some pretty big integers out there:

http://www.scottaaronson.com/blog/?p=2725

I think the current record is 1919 states, so if you run a certain Turing Machine for BB(1919) steps and it doesn't halt then you know that ZFC is consistent. Godelian considerations might make it reasonable to say that integers BB(1919) or larger don't exist.

I get skeptical of integers so large that you need weird Turing Machines to enumerate their digits whose halting proof is independent of set theory, but I don't yet disbelieve in their existence.

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