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

Ashley

@polyaletheia

Aug 12, 2020

I think there's a path between every pair of vertices in the empty graph?

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

)

@QiaochuYuan

Aug 12, 2020

this is the morally wrong definition of connectedness! you want connectedness to have the property that every graph has a unique (up to permutations) decomposition into connected components (analogous to prime factorization) and that's not true if the empty graph is connected

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

Replying to

and

my favorite definition of connectedness is "has exactly one connected component," where connected components are the equivalence classes of the equivalence relation generated by adjacency. the empty graph has zero. see also: ncatlab.org/nlab/show/too+

10:15 PM · Aug 12, 2020Twitter Web App

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