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Aug 12, 2020

thread: fun facts about zero / the empty set. i will die on all of these hills 1. 0^0 = 0! = {0 choose 0} = 1, these are all variations of the same fact which is that there's a unique function from the empty set to itself, which does nothing. this function is not constant

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Eliezer Yudkowsky

@ESYudkowsky

· Aug 12, 2020

How many different ways are there to choose 0 items from an empty set?

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166

Jay Daigle

@ProfJayDaigle

Aug 12, 2020

I don't really like calling this a "fact". It's a good definitional choice, but it really still is a _choice_.

QC (in Berlin 7/29-??? w/ COVID

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and

it's not a choice, that's part of what i mean when i say i'll die on these hills. if you break these facts then the category of sets loses some very basic and desirable properties

10:33 PM · Aug 12, 2020Twitter Web App

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and

That's what makes it a good choice!

QC (in Berlin 7/29-??? w/ COVID

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and

like you *could* define 0 + 0 = 1 if you really wanted to, that is in some sense a choice, but nobody would do that because then addition would fail to be associative. that's the same sense in which this isn't a choice, to a category theorist

Jordan Ellenberg

@JSEllenberg

Aug 12, 2020

Exactly: 0+0=1, as you say, is something no one would do, while "0^0 is undefined" is certainly something people do.

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and

As Jordan says, that's exactly what makes it a (good) choice! You start by saying "I'd like the category of sets to have these properties; how can I choose my definitions to give me those properties?" And this choice solves those problems and mostly doesn't cause any.

Jay Daigle

@ProfJayDaigle

Aug 12, 2020

You're not starting from well-defined consensus rules and deriving this fact; you're starting with a desired outcome, and choosing your axioms so you get it. Which is, like, half of doing math. And it's not a criticism, and this is the definition I use too in most circumstances.

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Jay Daigle

@ProfJayDaigle

Aug 12, 2020

It's axiomatic, sure. But axioms are _always_ choices. That's what makes them axioms and not theorems! You want the category of sets to have these nice properties, and so you choose axioms that make that happen. But if you had other goals, you could choose other axioms.

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