Mathematician with respect to the natural numbers: he is not concerned In this respect our position with respect to theĬharacter of the cardinals is similar to that of the working Whereas the cardinals are objects which we cannot even classify as sets The ordinals are particular, specified sets, There is, indeed, a crucial distinction between our construction of theĬardinals and the ordinals. The author has a few thoughts on this that I Operations on those from this axiom (and others), but we still don’t really know whatĪ cardinal number is. It’s interesting to see that we’re able to define cardinal numbers and deduce \(K(A) = K(B) \leftrightarrow A \approx B\) ![]() Formally, the axiom for cardinal numbers is: Notice that on the basis of this axiom and the otherĪxioms introduced we cannot prove that the cardinal number of a set is \(A\), such that with two equipollent sets we associate the sameĬardinal number. \(A\) is associated an object \(K(A)\), the cardinal number of Would like to follow Frege and Russell and define cardinal numbers asĮquivalence classes of equipollent sets, but we cannot prove that theĪppropriate equivalence classes exist. The intuitive idea of the axiom should be transparent. Namely, just that of the cardinal number of a set \(A\) (in symbols, The special axiom which we introduce requires a new primitive notion, Heĭoes this because he wants to avoid using the axiom of choice, and alsoīecause he says using the present axioms and the axiom of infinity toĭefine the rank of a set is rather complicated. Introduces a new temporary axiom in order to define cardinal numbers.
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