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I am self-studying set theory, following the book "The Joy of Sets". It says that an ordered pair $(a,b)$ is defined as the set $$\left\{\left\{a\right\},\left\{a,b\right\}\right\}.$$

Is this definition unique? Why cannot it be defined as $\left\{a,\left\{a,b\right\}\right\}$ or $\left\{a,\left\{b\right\}\right\}$?

Asaf Karagila
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Xenomorph
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  • It's not unique but it's common. – CyclotomicField Jan 07 '22 at 12:51
  • There are other definitions, this is by no means the only one. But you have to make sure that your definitions satisfy that $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$, and this should hold for any sets $a,b,c,d$. – Snaw Jan 07 '22 at 12:51
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    I suggest you take a look at https://en.wikipedia.org/wiki/Ordered_pair for more details. In short, there are multiple adequate variants and adequate definitions that are not variants of the above definitions. They all have advantages and disadvantages, though some have more disadvantages than others. Your first definition carries some major disadvantages explained by the article, which is why it is not preferrable. – Angel Jan 07 '22 at 12:58
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    Your second definition has the major disadvantage that it cannot distinguish between $({{a}},a)$ and $({a}, {a}),$ so it is simply not actually an adequate definition at all. – Angel Jan 07 '22 at 12:59
  • @Angel Thank you for your explanation. – Xenomorph Jan 07 '22 at 13:05
  • Your first definition does work. The proof for that definition that $(a,b)=(c,d) \implies (a=c) \land (b=d)$ uses the axiom of regularity; as the answer by Henno Brandsma points out, the traditional definition does not need that axiom. – aschepler Jan 07 '22 at 13:21

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Any definition of $(a,b)$ is fine as long as it satisfies

$$\forall a,b,c,d: (a,b)=(c,d)\iff (a = c) \land (b=d)$$

This definition, due to Kuratowski, is just the traditional one that we can define from just the pairing axiom (so with minimal axioms needed).

Others will do too, see Wikipedia for more.

Henno Brandsma
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  • It would have been useful to go into more detail as to why we accept the Kuratowski definition above the other definitions. Wikipedia partly explains this, but I gave further explanation as well in the comments. – Angel Jan 07 '22 at 13:00