Re: A question about ackermann's set theory.

On Tue, 30 Oct 2007 23:50:23 +0100, G. Frege <nomail@invalid> wrote:


Ackermann's set theory is present at

http://en.wikipedia.org/wiki/Ackermann_set_theory

One thing I don't understand is the reflection axiom schema

Concerning the comprehension schema they mentioned the following
restriction:

"[...] Let F(y, z_1, ..., z_n) be any formula which does not contain the
constant symbol V or the variable x free."

Maybe they forgot the mention a comparable restriction for the
reflection axiom schema (if so)?

I checked Ackermann's original paper (1956): yes, F(y, z_1, ..., z_n)
must be a formula which does not contain the constant symbol "V".

(Ackermann used a /predicate/ M, "is a set", instead. But "x e V" just
means that x is a set; since "V" is intended to denote the class of all
sets.)


My objection is the following:

Let F(y)<->(y e V & y is an ordinal)

This would violate the mentioned restriction (if so), since F(y) would
contain the symbol "V".

Indeed! (Well, it's Wikipedia, you know...)


Well done, Zuhair!


F.

--

E-mail: info<at>simple-line<dot>de
.

Relevant Pages

  • Re: A question about ackermanns set theory.
    ... Ackermann's set theory is present at ... One thing I don't understand is the reflection axiom schema ... Concerning the comprehension schema they mentioned the following ... constant symbol V or the variable x free." ...
    (sci.logic)
  • Re: A question about ackermanns set theory.
    ... reflection axiom schema? ... must be a formula which does not contain the constant symbol "V". ... Is the original paper something that you can send via e-mail. ... Well done, Zuhair! ...
    (sci.logic)
  • Re: A question about ackermanns set theory.
    ... One thing I don't understand is the reflection axiom schema ... My objection is the following: ... I don't know Ackermann set theory very well. ... you elaborate on exactly where the inconsistency comes in? ...
    (sci.logic)