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

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

)

@QiaochuYuan

Aug 12, 2020

2. the sum of zero terms (the empty sum) is 0; the product of zero terms (the empty product) is 1

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

)

@QiaochuYuan

Aug 12, 2020

3. the degree of the zero polynomial is -∞

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

)

@QiaochuYuan

Aug 12, 2020

4. the maximum of zero numbers is -∞; the minimum of zero numbers is +∞

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

)

@QiaochuYuan

Aug 12, 2020

5. the determinant of the 0x0 matrix (the empty matrix, with no entries) is 1

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

)

@QiaochuYuan

Aug 12, 2020

6. the empty graph is disconnected, and so is the empty topological space, for the same reason that 1 isn't a prime number

Wes Pegden

@WesPegden

Aug 12, 2020

I might call this one defensible, but certainly not a hill worth dying on. Note that the analogy with primality isn't clean; in particular, in this view, there's no analogous term for "composite" (≥2 components), which seems more natural a concept than "0 or ≥2 components".

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

)

@QiaochuYuan

Replying to

@WesPegden

this one has very practical consequences; there are various versions of the exponential formula which go something like (some sum over graphs) = exp(some sum over connected graphs) and they're false if the empty graph is taken to be connected

11:07 PM · Aug 12, 2020Twitter Web App