Count and Classification: Why Transfinite Cardinals Do Not Measure the Natural Numbers

A number of terms is the natural number at which a count of a collection termi-nates. An endless collection does not have a number of terms, because no count ofit terminates. A transfinite cardinal classifies sets by one-to-one mapping. Contraryto Cantor’s claim that the transfinite cardinals are numbers in the same sense as thefinite ones, a transfinite cardinal is not a number of terms. Cardinal and count agreeon finite sets only because the pairs of a finite mapping can themselves be counted.Galileo’s paradox rests on two different finite counts: squaring 1 through n yields nsquares, while only ⌊√ n⌋ squares lie among 1 through n. Cardinality extends the firstcount. Numerosity, under which a proper part is always smaller than its whole, extendsthe second. An answer to the question how many terms? must change when a term isadjoined or removed, and only then. Adjoining a member leaves a transfinite cardinalunchanged, while relabelling the members can change a numerosity. Cantor’s analogywith the irrational numbers fails: the diagonal, though incommensurable with the side,retains its length, while no number of terms remains when the count of an endless col-lection fails. An endless collection is nevertheless a determinate whole, against Leibniz.The set of real numbers has a greater cardinality than the set of natural numbers, yetthe statement that there are more real numbers than natural numbers is false as acomparison of numbers of terms. Every theorem of set theory stands.