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Topic: Torkel Franzen argues
Replies: 25   Last Post: May 17, 2013 3:52 PM

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Posts: 2,777
Registered: 12/13/04
Re: Torkel Franzen argues
Posted: Apr 26, 2013 1:09 PM
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On 26/04/2013 10:37 AM, Herman Rubin wrote:
> On 2013-04-25, FredJeffries <> wrote:
>> On Apr 25, 8:25 am, Alan Smaill <> wrote:
>>> Newberry <> 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"

> Mathematicians are willing to assume PA is consistent.

Agree. For the record I've always assumed PA is consistent,
until of course if one day ones present a proof _IN_ PA
of the form (F /\ ~F).

But assumption is _not_ assertion, wouldn't you agree?

(That's all I'm asking.)

> 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.

There is no remainder in the mathematics of infinity.


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