Date: Apr 25, 2013 5:51 PM
Subject: Re: Torkel Franzen argues
On Apr 25, 12:58 pm, FredJeffries <fredjeffr...@gmail.com> wrote:
> On Apr 25, 8:25 am, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> > Newberry <newberr...@gmail.com> writes:
> > > Torkel Franzen argues that all the axioms of ZFC are manifestly true
> > > the logic apparatus is truth preserving therefore all is good and the
> > > system is consistent.
> > Really??
> > Where did he make this claim?
> In "The Popular Impact of Gödel's Incompleteness Theorem"
> he says:
> "we can easily, indeed trivially, prove PA consistent using
> reasoning of a kind that mathematicians otherwise
> use without qualms in proving theorems of
> arithmetic. Basically, this easy consistency proof observes
> that all theorems of PA are derived by valid
> logical reasoning from basic principles true of the
> natural numbers, so no contradiction is derivable in PA"
Neither infinity as axiomatized in ZF nor regularity are obviously
manifestly true. Those restrictions of comprehension where the other
axioms simply expand comprehension don't necessarily reflect, for
example, any anti-foundational sets which some would have as obviously
Finite combinatorics and Presburger arithmetic are complete (where
unbounded and not necessarily infinite), regular axiomatization, or
rather, axiomatization as regular, of infinity, is disputable.