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From: Aatu Koskensilta on 11 Apr 2010 11:18
Newberry <newberryxy(a)gmail.com> writes:

> Such a proof in ZF that PA is consistent is obviously wothless.

Why? Are all proofs in ZF worthless?

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 11 Apr 2010 11:25
Marshall <marshall.spight(a)gmail.com> writes:

> I don't see any reason to pay much attention to anyone's
> intuition, my own included.

This is a sober attitude.

> "Intuition" is just a fancy word for "hunch."

"Intuition" can mean pretty much anything, from a vague hunch to
something very specific, as in e.g. Kant's thought. For example, when
people say that, say, the principle of mathematical induction is
intuitively evident it would be odd to take them to be telling us
they've a hunch it might hold.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 11 Apr 2010 11:28
Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> If you don't remember or have any doubt about my position, please
> allow me to clearly reiterate my position:
>
> It's impossible to have logically acceptable methods for proving a
> syntactical consistency.

But there's nothing clear about this reiteration. What is a "logically
acceptable method"?

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 11 Apr 2010 11:46
Aatu Koskensilta wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> If you don't remember or have any doubt about my position, please
>> allow me to clearly reiterate my position:
>>
>> It's impossible to have logically acceptable methods for proving a
>> syntactical consistency.
>
> But there's nothing clear about this reiteration. What is a "logically
> acceptable method"?
>

A method which would conform to definition of syntactical (in)consistency,
to definition of model of formal systems, to definition of rules of inferercne.
From: Nam Nguyen on 11 Apr 2010 11:53
Aatu Koskensilta wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> So what you're saying is you just _intuit_ PA system be consistent, no
>> more no less. Of course anyone else could intuit the other way too!
>
> Anyone can intuit whatever they want. I said nothing about such matters,
> I merely noted I have, like any number of logic students over the
> decades, produced a proof of the consistency of PA in the course of my
> studies. Intuition has no more and no less to do with it than with any
> proof in mathematics.
>

So, is your proof of the consistency of PA a syntactical proof of a
_consistent_ formal system? (I mean, anyone could syntactically prove
such proof in an inconsistent formal system). If not then in what sense
would your proof not require an _intuition about the natural numbers_?
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