[atwood22]

Do you know the definition of countable? A set S is countable if there is a one-to-one mapping from S to N where N is the natural numbers. Do you know that 0 is not a member of the natural numbers? We literally start counting at 1 by definition of countable.

[lupire]

Nope.

Is the empty set countable? (Yes.)

Dictionary:

nat·u·ral num·bers

  the positive integers (whole numbers) 1, 2, 3, etc., and sometimes zero as well

Countable: https://en.wikipedia.org/wiki/Countable_set

Set theory:

https://en.wikipedia.org/wiki/Ordinal_number

[atwood22]

From your own link on countable sets:

> Equivalently, a set S is countable if there exists an injective function f : S → N from S to N; it simply means that every element in S corresponds to a different element in N.

Defining N is usually done via a successor set, on which case 0 makes no sense to include.

[lupire]

A successor set is the set of successors of... 0 or 1, depending on what you are doing.

[kazinator]

An empty set is countable; it has an empty mapping to the natural numbers. Its cardinality is zero.

[lupire]

And its ordinalitiy is also 0.

Standard construction of ordinals is that each ordinal is the set of all its predecessors. (0 has no predecessors , hence 0 is the empty set.) (And so finite ordinals have the same ordinaliity as cardinality).

[pwdisswordfish9]

Show me a mathematical text where ‘ordinality’ is defined.

[samatman]

Are you a time traveller from the late 17th century?