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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: May 17, 2013 3:52 PM
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On 15/05/2013 11:21 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>> On 08/05/2013 8:11 AM, Nam Nguyen wrote:
>>> On 08/05/2013 7:28 AM, Frederick Williams wrote:
>>>> Nam Nguyen wrote:
>>>>> On 05/05/2013 8:45 AM, Frederick Williams wrote:

>>>>>> Nam Nguyen wrote:
>>>>>>> On 04/05/2013 10:07 AM, Frederick Williams wrote:

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

>>>>>>>>>> On 2013-04-25, FredJeffries <> wrote:
>>>>>>>>>>> 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,

>>>>>>>> How does one reach an infinite regression?

>>>>>>> By claiming that the state of consistency of PA can be
>>>>>>> proved _IN_ a _different formal system_ .

>>>>>> Your notion of infinite is very modest if does not go beyond two.

>>>>> That does _not_ mean there be only two, actually.

>>>>>>>>> and at that point
>>>>>>>>> we still have to confront with the issue of mathematical
>>>>>>>>> relativity.

>>>>>>>> It is not the case that either we go round in a circle or we
>>>>>>>> regress
>>>>>>>> forever.

>>>>>>> That's not a refute. Of course.
>>>>>>> (It's just an unsubstantiated claim).

>>>>>> And yet an obviously true one. Suppose the question of the
>>>>>> consistency
>>>>>> of PA is raised, a party to the discussion may say 'I accept that
>>>>>> PA is
>>>>>> consistent and I feel no need to prove it.' No circle, no
>>>>>> regression.

>>>>> The circularity rests with the argument on the _actual and objective_
>>>>> state of consistency of PA, _not_ on the _wishful and subjective_
>>>>> "acceptance" of anything.

>>>> Mathematicians (like the rest of humanity) are forever accepting
>>>> things. It is no big deal.

>>> Verification, proving, is a big deal.
>> For example, would you _accept_ the consistency of PA + ~cGC
>> ("It is no big deal" you said)?

> Not everyone shares your obsessions.
> The consistency of PA may be an objective fact (or fiction), but proving
> is a human activity.

Well then prove to the fora that cGC is true in the naturals, or that
~cGC is.

Isn't it true that Torkel Franzen once alluded to a similar notion that
one could "prove" the consistency of PA simply by observing that PA
wouldn't prove the false statement '0=1'?

But where would the false statement here: cGC, or ~cGC?

There is no remainder in the mathematics of infinity.


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