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In Axiomatic Set Theory Patrick Suppes introduces a new temporary axiom in order to define cardinal numbers. He does this because he wants to avoid using the axiom of choice, and also because he says using the present axioms and the axiom of infinity to define the *rank* of a set is rather complicated. I quote:

*The special axiom which we introduce requires a new primitive notion, namely, just that of the cardinal number of a set \(A\) (in symbols, \(K(A)\)). The intuitive idea of the axiom should be transparent. We would like to follow Frege and Russell and define cardinal numbers as equivalence classes of equipollent sets, but we cannot prove that the appropriate equivalence classes exist. So we postulate that with each set \(A\) is associated an object \(K(A)\), the cardinal number of \(A\), such that with two equipollent sets we associate the same cardinal number. Notice that on the basis of this axiom and the other axioms introduced we cannot prove that the cardinal number of a set is itself a set. Formally, the axiom for cardinal numbers is:*

\(K(A) = K(B) \leftrightarrow A \approx B\)

It’s interesting to see that we’re able to define cardinal numbers and deduce operations on those from this axiom (and others), but we still don’t really know what a cardinal number is. The author has a few thoughts on this that I quote:

*There is, indeed, a crucial distinction between our construction of the cardinals and the ordinals. The ordinals are particular, specified sets, whereas the cardinals are objects which we cannot even classify as sets or individuals. In this respect our position with respect to the character of the cardinals is similar to that of the working mathematician with respect to the natural numbers: he is not concerned with an explicit definition of the natural numbers in terms of other known entities, but only with knowning their essential mathematical properties. In particular, he requires that the natural numbers satisfy Peano’s five axioms, which may be formulated in elementary logic independent of set theory. These axioms are based on three primitive symbols: the predicate \(\text{is a natural number}\) the unary operation symbol \(\text{‘}\) for the successor function (intuitively \(x’ = x + 1\)), and the individual constant \(0\) for the number zero. Peano’s axioms are then:*

- \(0\) is a natural number.
- If \(x\) is a natural number then \(x’\) is a natural number.
- There is no natural number \(x\) such that \(x’ = 0\).
- If \(x\) and \(y\) are natural numbers and \(x’ = y’\) then \(x = y\)
- If \(\phi(0)\) and for every natural number \(x\) if \(\phi(x)\) then \(\phi(x’)\), then for every natural number \(x\), \(\phi(x)\).

*It is a proper question and not an idle philosophical speculation to ask why mathematicians agree almost uniformly on Peano’s axioms. Deep and difficult theorems about the natural numbers were proved before any adequate axioms were formulated. Putative axioms which did not yield these theorems would be rejected because, it seems, independent of any axioms there is a quite precise notion of what is true or false of the natural numbers. The author is not prepared to give any exact account of these intuitive notions, and it would be too much of a digression to examine what other people have said about these matters. But it should be realized that to say that the natural numbers* **are** *the finite cardinals or the finite ordinals is not to give such an exact account, for these intuitive yet precise ideas about the natural numbers are themselves used in deciding if the proposed identification is acceptable.*

But are Peano’s axioms actually defining natural numbers? Here are some thoughts Bertrand Russell had on Introduction to Mathematical Philosophy about this:

*In the first place, Peano’s three primitive ideas - namely “\(0\)”, “\(number\)” and “\(successor\)” - are capable of an infinite number of different interpretations, all of which will satisfy the five primitive propositions. We will give some examples.*

*Let “\(0\)” be taken to mean \(100\), and let “\(number\)” be taken to mean the numbers from \(100\) onward in the series of natural numbers. Then all our primitive propositions are satisfied, even the fourth, for though \(100\) is the successor of \(99\), \(99\) is not a “\(number\)” in the sense which we are now giving to the word “\(number\)”. It is obvious that any number may be substituted for \(100\) in this example.**Let “\(0\)” have its usual meaning, but let “\(number\)” mean what we usually call “even numbers”, and let the “\(successor\)” of a number be what results from adding two to it. Then “\(1\)” will stand for number two, “\(2\)” will stand for the number four, and so on; the series of “\(numbers\)” now will be \(0, two, four, six, eight, …\) All Peano’s five premises are satisfied still.**Let “\(0\)” mean the number \(1\), let “\(number\)” mean the set \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, …\) and let “\(successor\)” mean “half”. Then all Peano’s five axioms will be true of this set.*

*It is clear that such examples might be multiplied indefinitely. In fact, given a series \(x_0, x_1, x_2, x_3, …, x_n, …\) which is endless, contains no repetitions, has a beginning and has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano’s axioms. This is easily seen, though the formal proof is somewhat long. (…)*

*A series of [this] form (…) is called a progression.*