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I’m confused about the definition of a cardinal number.

We say the cardinal of a set $A$ is just the minimum element of the set of ordinals equinumerous to $A$.

But if two ordinals are equinumerous to $A$ does that not make the ordinals the same since every ordinal is the successor of the previous?

Also if the ordinals equinumerous to $A$ are all the same size then why do we pick the minimum in particular to be the cardinal of $A$ and how does this miminum ordinal actually correspond to the size/cardinal of the set $A$?

Partey5
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  • You keep talking about "size". – Angina Seng Apr 03 '20 at 21:51
  • Could you edit your penultimate paragraph's grammar to make its question clearer? Choosing the minimum equinumerous ordinal, or equivalently the intersection of such ordinals, is very natural; see also this concept, which doesn't require all sets to be equinumerous to an ordinal (a claim equivalent in ZF to the axiom of choice). – J.G. Apr 03 '20 at 21:52
  • I always thought the cardinal number of a set was the number of elements, wth care needed for infinite sets? – herb steinberg Apr 03 '20 at 21:54
  • @herbsteinberg In first-order theories without urelements where everything is a set, we instead choose a set representative of sets being a certain size. – J.G. Apr 03 '20 at 22:02
  • @J.G. My math education did not include first order theories, only basic set theory (Bourbaki). – herb steinberg Apr 03 '20 at 22:25

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if two ordinals are equinumerous to $A$ does that not make the ordinals the same since every ordinal is the successor of the previous?

No. Both $\omega$ and $\omega + 1 = \omega \cup \{\omega\}$ are equinumerous to $\omega$, but they are not the same.

if the ordinals equinumerous to $A$ are all the same size then why do we pick the minimum in particular to be the cardinal of $A$

Because it's convenient and always exists. There isn't a second-minimum ordinal equinumerous to $7$, for instance. On the other hand, there isn't a maximum ordinal equinumerous to $\omega$ (all ordinals below $\omega_1$ are equinumerous to it). So what other choice do you propose?

how does this miminum ordinal actually correspond to the size/cardinal of the set $A$?

It's the smallest ordinal that has the same size as $A$. That's it.

user76284
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