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This question may appear banal, but there seems to be more than meets the eye; a common glitch is to explain numbers by the "size" of sets without saying how to measure or compare the size of sets.

Can numbers be defined via Set theory alone, i.e. using only operations on sets or is it inevitable to use additional operations (careful here with equality of sets, because the cardinality could be the same, but the kind of elements may be different)?

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Ordinals are defined via set theory alone, finiteness is defined via set theory alone, and natural numbers are defined to be finite ordinals. No counting nor size measurement is involved in these definitions. –  Lee Mosher Mar 29 '13 at 13:39
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Once you've defined ordinals, you don't even need to introduce finiteness in order to define the natural numbers. They are the ordinals $x$ such that $x$ and all of its predecessors are successor ordinals or 0. –  Andreas Blass Mar 29 '13 at 13:42
    
Jacques Carette asked a variant of this question a while ago - mathoverflow.net/questions/17483/… –  François G. Dorais Mar 29 '13 at 15:24
    
Introducing the ordinals in the way that Toink outlined in his answer may be acceptable, but are finite ordinals really the correct analogue of natural numbers? Natural numbers are used for counting as well and thus could also be cardinal numbers. What I am still missing is an explanation how the definition of finite ordinal numbers relates to finite cardinal numbers, so that one can count the number of elements in a finite set. –  Manfred Weis Mar 29 '13 at 15:37
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@Manfred: you might get better, less terse, and more in-depth responses if you pursued this line of questioning on another site such as math.stackexchange.com. Your question, the terse answers of myself and Andreas, and the followup questions that you raise, are covered in elementary set theory texts, and as such are off topic for MathOverflow, which is intended for research questions. I have voted to close. –  Lee Mosher Mar 29 '13 at 16:00
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closed as not a real question by Lee Mosher, Steven Landsburg, Andreas Blass, Douglas Zare, DamienC Mar 30 '13 at 7:03

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2 Answers

up vote 3 down vote accepted

You can use Tarski's definition of finiteness:

$A$ is finite if and only if whenever a non-empty $U\subseteq\mathcal P(A)$ then there is some $A'\in U$ which is $\subseteq$-maximal element in $U$.

We also recall that a set $A$ is transitive if whenever $x\in A$ and $y\in x$, then $y\in A$.

Now you can say that $x$ is a natural number if it is a transitive set which is linearly ordered by $\in$ and finite. $\omega$ is the set of all natural numbers.

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You should probably also require $x$ to be transitive; otherwise $\{1\}$ satisfies your definition of natural number. –  Andreas Blass Mar 29 '13 at 17:38
    
@Andreas, yes. I meant that when I wrote $\in$ linearly orders $x$. But it might not be clear. –  Asaf Karagila Mar 29 '13 at 19:29
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Work in ZF(C)

Define $\omega := \bigcap${ $x\mid\emptyset\in x\wedge \forall y\in x\colon (y\cup$ {$y$}$)\in x$ }

By the axiom of infinity and of separation this is a set.

Now define the successor operation $s\colon \omega \to\omega$ by $y\mapsto y\cup$ {$y$} and $0:=\emptyset$. Then $\omega$ is the set of natural numbers and $1:=s(0)$, $2:=s(s(0))$, etc.

Of course this is just an ad hoc construction giving the same as defining all ordinals first (as transitive sets containing only transitive elements) and then doing what Andreas says

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The answers of Lee and Andreas leave the question open, how the ordinals are defined with set theoretic operations alone. Toink's construction is a definition by example, which I think, can be attributed to John von Neumann. My problem with that definition by example is, that it leaves open, how the so defined ordinal numbers relates to other sets. What would be the ordinal number of $\{red,green,blue\}$? if it should be 3 then how does that relate to s(s(s(0)))? –  Manfred Weis Mar 29 '13 at 15:21
    
As Lee pointed out in his comment, and as I therefore presupposed in mine, ordinals are defined in set theory alone. In ZFC they can be defined as transitive sets of transitive sets. Also, as Lee wrote in a later comment, things like this are off-topic for MO because they are covered in standard textbooks. –  Andreas Blass Mar 29 '13 at 17:41
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