About
> If "being" means "existence" then G"odel's completeness theorem suggests
> that it's not so absurd.
Cantor wrote in his 1883 *Grundlagen*, Section 8
"Mathematics is in its development entirely free and is only bound in the
self-evident respect that its concepts must both be consistent with each
other and also stand in exact relationships, ordered by definitions, to
those concepts which have previously been introduced and are already at
hand and established."
In taking 'consistency' to be a criterion for the immanent (mathematical)
existence of the transfinite numbers, I don't believe that he would have
settled for the existence of a countable model of some set of axioms for
them. He was arguing that the numbers themselves, uncountably many of them,
exist in the mathematical sense.
It is G\"odel's *in*completeness theorem that counts here: the second-order
theory of Cantor's number classes has no countable model (where by 'model'
I mean standard as opposed to arbitrary Henkin models).
But, I am also not happy with Robert T's
> "Consistency" carries with it a family of notions [and, as always when
> one word is chosen for a complex of inter-related notions, no one of
> which is central, it cannot be anything like simply defined].
> It carries with it the associations: coherent, of a piece (a unified
> if not complete whole), homogenous, standing-together, standing-firm,
> senseful, staying true to something, staying true...
I think that Cantor did mean freedom from contradiction by 'consistent with
each other'. Robert is certainly right that he would not have been
satisfied with the syntactical consistency of a formal system; and maybe
Robert's "staying true" is a reference to this.
I should mention that in his 1930 paper on set theory, Zermelo again refers
to consistency as a criterion for mathematical (ideal) existence---and he
clearly has in mind freedom from contradiction.
Dedekind, in his 1890 letter to Keferstein, explains that the point of his
construction of the Dedekind infinite system \<objects of my thought, my
ego, my thought of x\> was to ensure the consistency of the axioms of
number theory. Here again it seems to me that it is freedom of
contradiction that he had in mind.
Best regards,
Bill