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

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Frederick Williams

Posts: 2,164
Registered: 10/4/10
Re: Torkel Franzen argues
Posted: May 15, 2013 1:21 PM
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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.


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