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

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

)

@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)

Antoine Chambert-Loir

@achambertloir

Aug 12, 2020

That's an unreasonable convention. The dimension of a manifold is a locally constant function on the manifold.

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

)

@QiaochuYuan

Replying to

@achambertloir

you need a different empty manifold in each dimension for it to always be true that an n-manifold with boundary has a boundary which is an (n-1)-manifold, and in turn you need this to always be true to make cobordism categories work ncatlab.org/nlab/show/cobo

10:59 PM · Aug 12, 2020Twitter Web App

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

Replying to

and

one way to say this is that "n-manifold" fundamentally includes n as part of the definition; the fact that invariance of domain lets you recover n from the underlying topological space for non-empty manifolds is a convenience but it obscures what's going on

Akiva Weinberger

@akivaw

Aug 13, 2020

Is the disjoint union of two differently-dimensioned manifolds a manifold

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Antoine Chambert-Loir

@achambertloir

Aug 12, 2020

Replying to

@QiaochuYuan

The empty manifold is n-dimensional at each of its points, whatever n, I can't see your point.

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

)

@QiaochuYuan

Aug 12, 2020

i'm saying that a manifold is, conceptually speaking, more data than its underlying topological space, and the fact that invariance of domain lets you recover the dimension of a non-empty manifold from its underlying topological space is actually an accident

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