On 12/2/2013 8:34 PM, Zeit Geist wrote: > On Monday, December 2, 2013 1:58:01 PM UTC-7, WM wrote: >> >> >> Fine, then undefinable numbers cannot result from any Cantor-argument - and other theories do not contain them. So, why the heck do you think undefinable numbers exist? >> > > Although quite useful, not all of Set Theory depends on the Diagonal Argument. > > In fact, the existence or non-existence of non-constructible Real Numbers is Independent from the Axioms of ZFC. > > Some Models contain them, and some don't >
To the extent that WM's statements may be compared with typical mathematics, he does not speak of constructible reals.
One of the properties of terms introduced using definite descriptions is that they are eliminable from the formal language when the logic is first-order logic with identity.
The reason they are eliminable is because the definiens which expresses the properties that uniquely identify the object may be proven to identify no more than one object.
To the best that I can tell, this corresponds with Cohen's minimal model. Such a model is discussed in Mary Tiles' book. But, she does not identify it with Cohen's model.
Where Cohen discusses his model in terms of a cumulative hierarchy, it explicitly does not include a reflexive subset relation for the purpose of hierarchical construction. So, no object enters the model implicitly by virtue of the power set operation.
That would suggest that the only meaningful objects in the model are those which can be correlated in some way with descriptions.
But, I could be wrong. I have no verification from any outside sources.