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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

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@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

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@QiaochuYuan

Aug 12, 2020

3. the degree of the zero polynomial is -∞

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@QiaochuYuan

Aug 12, 2020

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

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

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@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

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@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

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

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@QiaochuYuan

Aug 12, 2020

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

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@QiaochuYuan

nah, if you want to assign the empty space a topological dimension it should be -∞ just like the degree of the empty polynomial. among other things this maintains additivity under product

11:03 PM · Aug 12, 2020Twitter Web App

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

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@QiaochuYuan

Aug 12, 2020

not familiar with the inductive dimension. according to wikipedia the inductive definition of the empty set is -1 and i guess you need that to make the induction to points work out. there is a statement along these lines i'll defend, which is that the empty set is the (-1)-sphere

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But he’s right that the n-dimensional empty manifold is just the empty space with an empty atlas of subsets of R^n?

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

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@QiaochuYuan

Aug 12, 2020

that's how i'd think about it anyway 👍