Re: Tarski's weird definition of cardinal numbers

Rupert wrote:
> > Thus all cardinal numbers are finite classes,
>
> Why? What about w={0,1,2,...}?
I meant finite numbers (representable by strings of digits) are finite
classes. I was ignoring omega and friends.

> You might
> have been thinking of the definition of ordinals as transitive sets
> well-ordered by the membership relation
Yes, that's what I meant, the one that says 0={}, 1={0}, 2={1,0}, etc,
except that I was (apparently mistakenly) calling them cardinal
numbers.
But why does this definition define ordinals and not cardinals?

> and cardinals as ordinals not
> equinumerous with any smaller ordinal; that's the usual definition
> today.
I understand the difference between using a number as a cardinal
(specifying quantity) and using it as an ordinal (specifying position),
but I don't understand the meaning of "cardinal number" and "ordinal
number", much less the difference between them, so I don't understand
the rationale behind their set-theoretic definitions. (And related to
this, are natural numbers cardinal numbers or ordinal numbers? I
thought they were simply numbers.)

> Then Tarski's definition has the advantage over this definition
> that you don't need to assume the axiom of choice to prove that any set
> has a cardinal
That only applies to sets with transfinite cardinality, right?

.

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