FINITIST

Bertrand Russell was considering a paradox found by Cantor. Cantor proved that the set of all subclasses of a set had more members of the class, so there is no largest class; but on the other hand, the class of all things altogether should ba as large as any class - nothing could be larger. In considering this Russell found an even simpler paradox. Consider the class of all classes that are not members of themselves. If it is a member of itself, then by definition, it is not a member of itself; and vice versa.

There are also other difficulties with infinity. I think we should, at least consider the possibilty that the number of things in the world is finite. Whitehead and Russell in Principia Mathematica, adduced an "axiom" [not really an axiom] of infinity, wherever they used it. I think it would be of interest to see what follows from assuming its negation.

I think its assumption useful for mathematics, but that, though useful, it is false. There is no actual infinite.  

I think PM is correct [philosophically] in taking intensions as prior to extensions. But, I believe intensions are psychological. It seems if we have an idea of 1 and an idea of n + 1, then we have an idea of any number reached by applying n + 1 to 1. But, actually, we have a finite limit to our capability. Also, intensions being psychological, are different for different people. It is OK, to see what can follow from various axiom systems. But I think these systems are seductive idealizations and not actual. Some can be useful - as in physics. Others, such as, at least parts of what is called "measure theory", I see no practical use for.

I think, for mathematics, it is sufficient to use extensions, and accept infinity as an [unreal] idealization.