An intutive counterexample to the axiom of choice

Re:An intutive counterexample to the axiom of choice by phkahler on Monday January 23, 2023 @09:07AM (#34488837)

We need to switch to the scheme that converts name/description to an integer for that. If you mean the repeating decimal, I don't know if it's before or after pi because I can't identify (name) their locations.

It's the same argument. If I can accept that there are more reals than reals with finite descriptions, why can't there be integers without finite descriptions/positions?

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Re:An intutive counterexample to the axiom of choice by jacobmartin on Monday January 23, 2023 @09:15AM (#34488945)

> If I can accept that there are more reals than reals with finite descriptions, why can't there be integers without finite descriptions/positions?

Because the integers are defined as the finite successors to the number 0 and their additive opposites. See, for instance, the construction section of: https://en.m.wikipedia.org/wiki/Integer

You can start talking about numbers that can't be constructed in this way, but then you've ceased to be talking about the integers.

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Re:An intutive counterexample to the axiom of choice by phkahler on Monday January 23, 2023 @09:25AM (#34489074)

Is the size of the set of integers an integer?

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Re:An intutive counterexample to the axiom of choice by jacobmartin on Monday January 23, 2023 @09:28AM (#34489115)

No, usually the cardinalities of infinite sets are represented by things called aleph numbers: https://en.m.wikipedia.org/wiki/Aleph_number.

But it isn't, in any case, an integer.

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Re:An intutive counterexample to the axiom of choice by phkahler on Tuesday January 24, 2023 @11:02AM (#34505315)

Thanks! That to me is the most useful response to my ramblings today!

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