[FOM] Platonism 1

[William Tait]

I have been slowly absorbing the contributions to the recent thread(s) 
on Platonism and formalism. They raise many interesting points for me 
and I want to see if I can sustain a series of comments on it. Hence 
the new subject heading.

I first want to comment on the view---that, to my embarrassment, I have 
held---that, whereas classical logic involving only bounded 
quantification over sets is justified, in the case of unbounded 
quantification over `all sets', the appropriate logic is 
intuitionistic, because of the open-ended character of this universe. 
This view depends upon taking the universe of `all sets’ to be the 
intended model of set theory.  This view is mentioned by A.P. Hazen 
(9/6/03), citing a paper by Poszgay.

A contrasting view, which goes back to Zermelo’s  ``Ueber Grenzzahlen 
und Mengenbereiche: Neue Untersuchungen ueber die Grundlagen der 
Mengenlehre’’ [Fundamenta Mathematicae 16 (1930): 29-47], is that the 
models of set theory form (to within isomorphism) an unbounded 
increasing sequence.  (By set theory, he means the impredicative 
second-order ZFC.)  These models are determined as the least epsilon 
structures satisfying specific consistent extensions of ZFC and thus 
have what for him is the essential property of a mathematical 
structure: a categorical axiomatic theory.  On this conception, the 
logic of unbounded quantifiers should be treated no differently from 
that of bounded quantifiers. For, applied to any model, they are 
quantifying over a  well-defined set M, which indeed is an element of 
each of the succeeding models.

In spite of incompleteness and the inability (in any relevant sense) to 
prove consistency (of which Zermelo was of course unaware then), this 
remains an attractive---even compelling---conception.  For it makes no 
presumption that there is a unique way in which set theory will 
develop. New axioms may indeed be seen as `true’, but not in the 
semantical sense that they hold in the universe of all sets (for then 
their truth is speculative only), but rather in the sense that they 
develop correctly some conception of a universe of sets. On this view, 
the only grounds for asserting truth in the semantical sense is proof 
from the axioms. The challenge is of course to arrive at a conception 
of a universe of sets on the basis of which the axioms are `true' in 
the former sense.

(I think that maybe this distinction between two senses of `true’ is 
related to a discussion in a recent posting of the distinction between 
`correspondence theories of truth’ and `coherence theories of truth’.)

The alternative I mentioned to Zermelo's position  seems to me to be an 
instance of a kind of Platonism, that Wang (or Goedel) called 
`conceptual Platonism’, according to which the concept of `set’ picks 
out by itself some well-determined extension. I can’t see that Goedel 
gave an argument for this kind of Platonism, although he seems to have 
subscribed to it. I also don't see what argument there can be for it. \

I want to distinguish this sense of Platonism from another, which 
Goedel did cogently argue for and which I would defend. But I will 
discuss this in another posting.

Bill Tait