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.