Re: Provable soundness of finite subtheories of ZFC

MoeBlee <jazzmobe@xxxxxxxxxxx> writes:

On Nov 28, 10:34 am, Aatu Koskensilta <aatu.koskensi...@xxxxxx> wrote:
As everyone knows, ZFC is reflexive, that is, proves the consistency
of each of its finite subtheories. More is true, however: ZFC proves
not only the consistency of each of its subtheories, but also their
soundness, in the form of the uniform reflection schema. In more
detail, it proves, for every finite subtheory A, all instances of

 (n)Prov_A("P(n)") --> (n)P(n)

where n is a variable ranging over the naturals. This can be
established easily by using reflection and a truth predicate applying
to sentences of bounded quantifier complexity. (It can of course also
be established proof-theoretically by an appeal to the Haupsatz, which
is provable in ZFC.)

I wonder whether the simple set-theoretic argument is spelled out in
detail in any of the standard texts on set theory?

By 'standard texts' do you mean basic undergraduate introductions to
set theory? I've not seen this matter discussed in any of those books
I've looked through. Maybe, though, in more advanced books such as
Kunen's or Jech's?

I had in mind such more advanced texts -- I doubt the matter would
come up in an elementary exposition. There is of course an easy way
for me to find out, by simple inspection, but I was hoping someone
would remember seeing the argument spelled out somewhere...

(By the way, I have the new revised edition of Jech on order, on the
way to me!)

A fine Christmas present. Perhaps when you receive the book you might
say a few words about what's new in the 2006 edition.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxx)

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

Relevant Pages

  • Provable soundness of finite subtheories of ZFC
    ... As everyone knows, ZFC is reflexive, that is, proves the consistency ... of each of its finite subtheories. ... in the form of the uniform reflection schema. ... established easily by using reflection and a truth predicate applying ...
    (sci.logic)
  • Re: Provable soundness of finite subtheories of ZFC
    ... of each of its finite subtheories. ... More is true, however: ZFC proves ... detail in any of the standard texts on set theory? ... Maybe, though, in more advanced books such as ...
    (sci.logic)
  • Re: Provable soundness of finite subtheories of ZFC
    ... of each of its finite subtheories. ... More is true, however: ZFC proves ... detail in any of the standard texts on set theory? ... Maybe, though, in more advanced books such as ...
    (sci.logic)
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