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Why in set Theory do we define cardinals to be sets and not numbers? Throughout university we were always told that the cardinality of a set is just the number of elements in the set but not when I study ZFC set Theory I find out a cardinal of a set is the minimum ordinal equinumerous to the set.

I can not grasp at all how these sets correspond to numbers? Why can’t we have numbers without making them be strange definitions involving sets?

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    $\begingroup$ Everything is a set in set theory, even the natural numbers are implemented as sets, so why shouldn't the cardinals also be? $\endgroup$ – Alessandro Codenotti Apr 3 '20 at 22:16
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    $\begingroup$ What exactly is an infinite number? They're undefined ... unless you define them somehow. Defining everything as a set, even numbers themselves, is the "machine language" of set theory. $\endgroup$ – runway44 Apr 3 '20 at 22:18
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    $\begingroup$ Set theory is a one sorted theory, meaning there is only one "type" of object: set. There are no numbers, functions, pairs or anything else, there are only sets, so all of this other "types" are implemented as sets. This is not the only possible approach to the foundations of maths, but it's how things are done in set theory $\endgroup$ – Alessandro Codenotti Apr 3 '20 at 22:24
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    $\begingroup$ In ZFC, everything is a set. This is because ZFC strives to be an example of a first-order logic. $\endgroup$ – Thomas Andrews Apr 3 '20 at 22:29
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    $\begingroup$ @Thomas Weird phrasing. (It is a first-order theory, there is not much striving going on.) $\endgroup$ – Andrés E. Caicedo Apr 3 '20 at 22:38
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What are numbers though?

Remember, from a set-theoretic perspective, everything has to be defined in terms of sets. More precisely, you have to "build" everything up from the empty set using the ZF(C) axioms.

This includes numbers!

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As Robert Israel points out in his answer, it's not really important that they are sets "conceptually", we still think of them as numbers, but we want to have a fundamental object which we can build everything out of, and in mathematics, that object is the set. This is analogous to how everything in a computer, when it comes down to it, is a sequence of 1's and 0's (every document, picture, video, etc.).

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  • $\begingroup$ That’s a great analogy! $\endgroup$ – Andrew23 Apr 3 '20 at 23:27
  • $\begingroup$ How does the definition of cardinals and ordinals allow us to think of them as we want to think of them I.e representing size and order of sets? $\endgroup$ – Andrew23 Apr 3 '20 at 23:45
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    $\begingroup$ @Anteater23 The idea is that we treat them as the "canonical" sets which define size. For instance, the cardinal $3$ defines what it is to have 3 elements. So, rather than saying that $\{a,b,c\}$ has size 3, we say "it has the same size as the cardinal $3$", which is actually a set containing $\{0,1,2\}$. $\endgroup$ – Luke Collins Apr 3 '20 at 23:48
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In general in mathematics, we're not really concerned with what something is, rather with what can be done with it. So if you want to define real numbers as Dedekind cuts, and I want to define them as equivalence classes of Cauchy sequences of rationals, we don't need to argue about it: there is a one-to-one correspondence between your "real numbers" and my "real numbers", which preserves all algebraic and analytic structures. It's the same with cardinal numbers. Set theorists want to define them as certain sets; maybe you want to define them some other way. It doesn't matter as long as you have a consistent definition which satisfies all the properties that cardinal numbers should have.

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