Conversation

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

)

@QiaochuYuan

Aug 12, 2020

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

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

Replying to

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

10:34 PM · Aug 12, 2020Twitter Web App

Replying to

and

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

Jay Daigle

@ProfJayDaigle

Aug 12, 2020

And I can come up with contexts where you want to say 0^0=0, as well. If f(x) = 0^x, then it's way more convenient to define f(0)=0 than f(0)=1.