Philosophy of Mathematics

by Øystein Linnebo

A cardinal number is supposed to be some form of property that two families share just in case they are equivalent in this way. So why not simply identify the cardinal number of a family with the clan consisting of all families equinumerous with this given family? This is mathematically elegant, and it ensures that two families have the same cardinal number just in case they are equinumerous. As a logicist account of mathematics, however, STT is now generally regarded as a failure. One problem is that the strict division into types is cumbersome and needlessly restrictive.3 A related problem is that its syntactic restrictions on membership claims block not only the paradoxes but also Frege’s celebrated bootstrapping argument, which shows that there are infinitely many numbers. The argument crucially turns on the fact that numbers too can be counted. Thus, if we have established the existence of numbers 0 through n, we have at least n + 1 objects available to count, which enables us to establish the existence of the number n + 1 as well. This argument is no longer available when numbers are construed in Russell and Whitehead’s way as clans of equinumerous families. As a result, we need a separate axiom of infinity, which states that there are infinitely many individuals. But it is doubtful that this axiom can be regarded as a purely logical principle. It certainly does not qualify as logical on our contemporary conception of logic, which requires a logical truth to be true ...