Re: Let's you & him fight

Rupert wrote:
Never mind all this talk about sets and classes. Suppose I adjoin to
the first-order language of set theory a satisfaction predicate, the
idea being that Sat(n,x_1,x_2,...,x_k) holds when n is the Goedel
number of a formula in the first-order language of set theory with k
free variables and it is true when these variables are interpreted to
be x_1, x_2, ... x_k. I adjoin all the obvious axioms concerning the
satisfaction predicate and I assume that the axiom of separation and
replacement hold for formulas involving Sat as well. The resulting
theory, call it ZFC*, is not conservative over ZFC, it has much
stronger consistency strength. But surely if we accept ZFC we should
accept ZFC*. And NBG can be interpreted in ZFC*.

Quite so. In fact set theoretical truth is definable in NBG: there is a formula True(x) such that NBG proves for every formula A that True('A') <--> A. NBG does not prove all the usual clauses in the usual inductive definition for truth, however. I gave an axiomatization of NBG based on this observation some time ago here in sci.logic.

And of course, from the observation that ZFC* is acceptable on basis of our picture of the world of sets, we see that ZFC** with a second truth predicate covering also formulas involving the truth predicate of ZFC* is also acceptable, and similarly for ZFC*** and so forth.

The situation is analogous to the analysis of classical predicativism - "predicativism given the natural numbers" as Feferman has called it. Informally the replacement axiom says

for any relation R, if for every x there is an unique y such that xRy
then for any set A the set {y | Ex in A such that xRy} exists

When we formalize this we need to explicate what sort of relations there are. As a first approximation we can take just the set theoretically definable relations, giving us ZFC. But because we recognize the intelligibility of the language of set theory, we can allow for relations definable in the language of set theory augmented with a truth predicate, and so forth. Working out the details we get a theory in which we postulate the existence of any relation definable by iterating the truth predicate along any - possibly class sized - well-ordering that becomes definable in the process. Now, this doesn't really make any sense: the totality of class sized well-orderings is not determinate; we can always come up with more and more such well-orderings. But if we adopt a "perspective from above" and accept e.g. the existence of an inaccessible kappa, we can talk about the well-orderings that become definable when the process is carried out in V_kappa, and we can see that the theory obtained is satisfied in <V_kappa, L[V_kappa]_alpha> where alpha is the limit of well-orderings obtainable autonomously, similarly as omega^CK_1 is the limit of predicatively definable countable ordinals. Thus, given we accept an inaccessible or something beyond ZFC, we can recognize the resulting theory as comprising exactly the principles that are acceptable on basis of acceptance of ZFC, similarly as RA_Gamma_0 - predicative analysis - comprises exactly the principles acceptable on basis of acceptance of the totality of natural numbers as determinate - so that arithmetical truth is a legitimate notion - and induction in the informal sense.

One often hears things like

Gödel's and Cohen's work shows that CH can't be decided by the
currently accepted set theoretical principles.

This, of course, is false. The work of Gödel and Cohen only shows that CH can't be decided in ZFC. But the statements isn't false in any dangerous sense - no innocent soul is led horribly astray - since it's trivial that Gödel's and Cohen's proofs go trough for much stronger theories than ZFC, e.g. ZFC + there is an inaccessible, which certainly include everything that is provable using the currently accepted set theoretical principles.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

Relevant Pages

  • Re: Skolems Paradox and why is math the way it is?
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  • Re: Small Set Theory,Updated.
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  • Re: provability and truth
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  • Re: Simplifying M theories.
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