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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 5:26 PM
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On 26/04/2013 11:09 AM, Nam Nguyen wrote:
> 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.

Exactly right. And exactly my point.

Somewhere, somehow, a circularity or an infinite regression
of _mathematical knowledge_ will be reached, and at that point
we still have to confront with the issue of mathematical relativity.

There's really no escape to it, I'm afraid from what I could gather.

There is no remainder in the mathematics of infinity.


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