Conversation

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

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

· Aug 12, 2020

How many different ways are there to choose 0 items from an empty set?

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

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

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

3. the degree of the zero polynomial is -∞

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

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

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

10:03 PM · Aug 12, 2020Twitter Web App

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

Replying to

7. there's a different empty manifold in each dimension: a zero-dimensional empty manifold, a 1-dimensional empty manifold, etc. (and of course none of them are connected or path-connected)

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

8. the zero-dimensional vector space (consisting only of the vector 0) has a unique basis, which is empty

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9. (this one is new to me, thanks

!) the zero endomorphism of the zero banach space has operator norm 0; the endomorphisms of the zero banach space do still form a unital banach algebra but only if you relax the unit axiom to |id| \le 1

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Antoine Chambert-Loir

· Aug 12, 2020

Replying to @achambertloir and @QiaochuYuan

By the way, for the operator norm of an endomorphism, the supremum is taken in R_+ adjoined +infinity, so that the norm of the identity endo of the null space is 0.

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

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

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

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Replying to

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".

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(But I made it this far in otherwise in complete agreement.)

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Adele Dewey-Lopez

@AdeleDeweyLopez

Replying to

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

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

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

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I'll grant you that "X is connected" should probably mean "the 0-truncation of X = the (-2)-truncation of X", so empty things are ruled out. But I think the notion "the 0-truncation of X = the (-1)-truncation of X" ought to have its own name too. "Connected-if-inhabited"?

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

some concept in this direction maybe deserves the name “subconnected”

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