On 2013-04-25, FredJeffries <email@example.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.
>> Where did he make this claim?
> In "The Popular Impact of Gödel's Incompleteness Theorem"
> "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"
Mathematicians are willing to assume PA is consistent. The inconsistency of PA would mean that the basic principles of the natural numbers are inconsistent.
I recommend that the discussion of the natural numbers from the basic principles be taught very early, and addition, etc., be derived from them. They LOOK obvious. But if it is consistent, we know we cannot prove it.
Now PA has been proved consistent in ZF or NBG, but then that brings the consistency of axioms for set theory.
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University firstname.lastname@example.org Phone: (765)494-6054 FAX: (765)494-0558