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

Adele Dewey-Lopez

@AdeleDeweyLopez

Aug 13, 2020

hmm, not sure i agree that it's for the same reason that 1 isn't prime... (ofc i agree with the statements)

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

)

@QiaochuYuan

Replying to

@AdeleDeweyLopez

how do you feel about the analogy between prime factorization and the decomposition of a graph into connected components?

12:47 AM · Aug 13, 2020Twitter Web App

Replying to

seems only loosely related: 1. multiple units can exist 2. prime factorization isn't necessarily unique even up to units 3. is there an analogue to the fundamental 0-groupoid (i.e. set) construction for the prime factorization context? (would be pretty awesome if there is tbh)

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

)

@QiaochuYuan

Aug 13, 2020

1. you can switch to working with ideals 2. only talking about the cases where it's unique in the usual sense, e.g. the integers 3. you can take the peirce spectrum of the affine scheme Spec Z/(n) lol

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