A Scrabble on Cantor, Hilbert, and Zermelo (2)
We must take a step back. Cantor claimed that if the forming process is consistent, the formed object exists. So, the problem of consistency is crucial in this argument. Cantor was very sure of existence of objects which his own mathematical theory involves: infinite sets, transfinite ordinals, etc. Moreover, he tried so hard to defend their existence against many criticisms -- especially Kronecker's. But as one can see, one of the curious things on Cantor is that he did not even try to show the consistency of his theory. No treatise about consistency.*1 Cantor said almost nothing about logic and metamathematics.
Now, if one accepts sometimes-called `axiomaticism' -- the objects can be thought only in the context of a theory,*2 the question on the existence of mathematical objects turns into one whether there exists any `model' that the theory tries to describe. If this model exists, we can infer the existence of mathematical objects, which are contained in the (domain of) model. Thus, Cantor's claim becomes like this:
A model that the theory tries to describe exists if that theory is consistent.
Ah, we all are familiar with this claim -- completeness. Cantor's resort to God changes into completeness through axiomaticism.
