Date: May 17, 2013 3:52 PM Author: namducnguyen Subject: Re: Torkel Franzen argues 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 <fredjeffries@gmail.com> 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.

NYOGEN SENZAKI

----------------------------------------------------