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 +∞

Antoine Chambert-Loir

@achambertloir

Aug 12, 2020

In some conventions, maximum and minimum are always attained — largest and smallest elements. Minus infinity is the least upper bound (in R adjoined plus/minus infinity) of the empty set.

Antoine Chambert-Loir

@achambertloir

Aug 12, 2020

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.

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

)

@QiaochuYuan

Aug 12, 2020

whoa interesting. trying to decide if this implies endomorphisms of the zero banach space isn't a banach algebra...

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

)

@QiaochuYuan

Aug 12, 2020

okay this is trippy thanks for this. the operator norm of the empty matrix is zero. it has no singular values so the max of its singular values is 0. but this means either endomorphisms of the zero banach space isn't a (unital) banach algebra, or i have to drop |id| = 1...

Antoine Chambert-Loir

@achambertloir

Aug 12, 2020

You have to ask that |id| <= 1. Or rather: The norm of an arbitrary finite product is smaller than the product of the norms.

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

)

@QiaochuYuan

Replying to

@achambertloir

very slick, thank you

11:33 PM · Aug 12, 2020Twitter Web App