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" > > http://www.ams.org/notices/200604/fea-franzen.pdf > > 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 existant.
Finite combinatorics and Presburger arithmetic are complete (where unbounded and not necessarily infinite), regular axiomatization, or rather, axiomatization as regular, of infinity, is disputable.