Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

From: Aatu Koskensilta (aatu.koskensilta_at_xortec.fi)
Date: 01/30/05 

Date: Sun, 30 Jan 2005 12:59:15 +0200

Torkel Franzen wrote:

> Second, even given this stipulation, you haven't actually defined any set
> for which it makes any sense to ask whether or not it is recursively
> enumerable. "ZFC plus reflection-type principles" is too vague for
> this.

There is a way of defining "ZFC plus reflection-type principles" which
basicly amounts to adding proper classes to ZFC with an axiom saying
that V exists, an axiom saying that every set is a class and a rule of
inference saying that if a class B is an ordinal, then for every class A
the class of predicatively definable classes relative to A up to B
exists. The replacement and separation schemes should naturally be
expressed as Pi^1 axioms. I don't know how this theory relates to
Feferman's reflective closure of ZFC or his unfolding of set theory.

Of course, there is no reason to suppose that "recognizable as true" is
the same thing as provability in "ZFC plus reflection-type principles".
For one, if one accepts the analysis of what is acceptable on basis of
ZFC outlined in the previous paragraph then the soundness of "ZFC plus
reflection-type principles" should be accepted as well. 

-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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