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I've been reading the following question about the universal property for an infinite cartesian product: Two Definitions of Infinite Cartesian Product

The thing is that the accepted answer proves that two sets verifying the universal property have bijections to each other, so this means, at least to my understanding, that both sets should have the same cardinality. Is this true? If so, that would mean that any set verifying the universal property have the very same cardinality (that is which one exactly?). I find this very strange, and I think I'm interpreting something wrong. Can you explain it to me?

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    $\begingroup$ Yes, if there is a bijection between two sets, then those two sets have the same cardinality (by definition). And so yes, all sets that satisfy a particular universal property will have the same cardinality. The purpose of a universal property is that it specifies exactly one set, up to bijections that preserve all the relevant structures. $\endgroup$ – Greg Martin Aug 31 at 19:21
  • $\begingroup$ That creates a problem for me. In the question I linked I think it's proved that any cartesian product of sets verify the universal property, so that would mean all cartesians product of sets have the same cardinality, which it's clearly not true. $\endgroup$ – Darsen Aug 31 at 19:30
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    $\begingroup$ Any cartesian product of particular sets satisfies the universal property for those sets. The universal property satisfied by $\{1\}\times\{1\}$ is not the same as the universal property satisfied by $\Bbb R\times\Bbb R$. $\endgroup$ – Greg Martin Aug 31 at 19:31
  • $\begingroup$ Oh, I see. Then given any collection of sets $\{X_i\}_{i \in I}$, a set $Q$ with a family of maps $\{q_i : Q \to X_i\}_{i \in I}$ that verifies the universal property has the same cardinality as the set $\prod_{i \in I}X_i$, am I right? $\endgroup$ – Darsen Aug 31 at 19:56
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    $\begingroup$ Exactly. (I would say it satisfies the universal property, rather than verifies it.) $\endgroup$ – Greg Martin Aug 31 at 20:15
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Yes, if there is a bijection between two sets, then those two sets have the same cardinality (by definition). And so yes, all sets that satisfy a particular universal property will have the same cardinality. The purpose of a universal property is that it specifies exactly one set, up to bijections that preserve all the relevant structures.

Any cartesian product of particular sets satisfies the universal property for those sets. The universal property satisfied by $\{1\}\times\{1\}$ is not the same as the universal property satisfied by $\Bbb R\times\Bbb R$, for example.

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