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

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Registered: 12/13/04
Re: Torkel Franzen argues
Posted: May 8, 2013 11:44 PM
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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)?

There is no remainder in the mathematics of infinity.


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