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

Owen Biesel

@owenbiesel

Aug 12, 2020

Did a double take at "[the unique function from the empty set to itself] is not constant", but of course you're right! It's only locally constant.

Lynn

@chordbug

Aug 12, 2020

I suppose for *most* functions X → Y, the definitions ∃y: ∀x: f(x)=y and ∀x: ∀x': f(x)=f(x') are equivalent, but the ∅→∅ function is “vacuously constant” by the second and “trivially non-constant” by the first… is one of these definitions vastly better/more “natural”?

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

Replying to

and

a clean categorical definition of "constant" is "factors through 1" and the empty function 0 -> 0 doesn't said a simpler way, and similar to "a connected graph has exactly 1 connected component," a constant function is a function that takes exactly 1 value

11:41 PM · Aug 12, 2020Twitter Web App

Replying to

and

wow, it seems even category theorists have Takes about this ncatlab.org/nlab/show/cons (“See the forum for further discussion of this.”)

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

)

@QiaochuYuan

Aug 12, 2020

hmm i think these definitions are just two different concepts and the first one should be called "subconstant"