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From: Nam Nguyen on 3 Jul 2010 10:25
herbzet wrote:
>
> Nam Nguyen wrote:
>> herbzet wrote:
>>> Nam Nguyen wrote:
>>>> herbzet wrote:
>>>>> The whole thing is nonsense, anyway. Clearly, PA
>>>>> is consistent, or at least, its consistency is
>>>>> at least as evident as the consistency of any
>>>>> system which purports to prove it.
>>>> Sometimes it's much ... much simpler and more logical, humble,
>>>> humanistic to admit we don't know what we can't know, rather
>>>> than pretending to possess some sort of an immortal knowledge.
>>>>
>>>> Suppose someone states "There are infinitely many universes
>>>> and each has harbored a planet with intelligent life in its history."
>>>>
>>>> If there actually are infinitely many universes we can't know such
>>>> fact. PA's consistency is like such a statement: if it's consistent,
>>>> you can't never know that. Period.
>>> I see a model for PA: the natural numbers. I conclude PA is consistent.
>>>
>>> I would say I know this to a mathematical certainty.
>> While having such a "seeing", do you "see" if PA + (1) is consistent?
>>
>> I mean, how far can you go with such "seeing"?
>
> What is PA + (1)?

Let's define the 2 formulas in L(PA):

pGC <-> "There are infinitely many examples of Goldbach Conjecture."
cGC <-> "There are infinitely many counter examples of Goldbach Conjecture."

Then:

(1) <-> pGC xor cGC
From: Nam Nguyen on 3 Jul 2010 10:35
herbzet wrote:
>
> Nam Nguyen wrote:
>> herbzet wrote:
>>> Nam Nguyen wrote:
>>>> herbzet wrote:
>>>>> The whole thing is nonsense, anyway. Clearly, PA
>>>>> is consistent, or at least, its consistency is
>>>>> at least as evident as the consistency of any
>>>>> system which purports to prove it.
>>>> Sometimes it's much ... much simpler and more logical, humble,
>>>> humanistic to admit we don't know what we can't know, rather
>>>> than pretending to possess some sort of an immortal knowledge.
>>>>
>>>> Suppose someone states "There are infinitely many universes
>>>> and each has harbored a planet with intelligent life in its history."
>>>>
>>>> If there actually are infinitely many universes we can't know such
>>>> fact. PA's consistency is like such a statement: if it's consistent,
>>>> you can't never know that. Period.
>>> I see a model for PA: the natural numbers. I conclude PA is consistent.
>>>
>>> I would say I know this to a mathematical certainty.
>> IOW, mathematical reasoning is just a gambling that never ends!
>
> You find something doubtful in the proof of the infinitude of primes?

There are a lot of things I never find doubtful, such as the proof
Ax[~(Sx=0)] in PA. But there are also a lot of theories in which not
only Ax[~(Sx=0)] is provable but also Axy[x*y=0] is. By the natural
numbers, would you mean they collectively be a model of such theory?

Just _exactly_ what do you "see" as the natural numbers? If there's
a least one formula you couldn't its truth in the naturals, in what
sense do you "know" the naturals?
From: herbzet on 3 Jul 2010 12:42


Nam Nguyen wrote:
> herbzet wrote:
> > Nam Nguyen wrote:
> >> herbzet wrote:
> >>> Nam Nguyen wrote:
> >>>> herbzet wrote:

> >>>>> The whole thing is nonsense, anyway. Clearly, PA
> >>>>> is consistent, or at least, its consistency is
> >>>>> at least as evident as the consistency of any
> >>>>> system which purports to prove it.
> >>>>
> >>>> Sometimes it's much ... much simpler and more logical, humble,
> >>>> humanistic to admit we don't know what we can't know, rather
> >>>> than pretending to possess some sort of an immortal knowledge.
> >>>>
> >>>> Suppose someone states "There are infinitely many universes
> >>>> and each has harbored a planet with intelligent life in its history."
> >>>>
> >>>> If there actually are infinitely many universes we can't know such
> >>>> fact. PA's consistency is like such a statement: if it's consistent,
> >>>> you can't never know that. Period.
> >>>
> >>> I see a model for PA: the natural numbers. I conclude PA is consistent.
> >>>
> >>> I would say I know this to a mathematical certainty.
> >>
> >> While having such a "seeing", do you "see" if PA + (1) is consistent?
> >>
> >> I mean, how far can you go with such "seeing"?
> >
> > What is PA + (1)?
>
> Let's define the 2 formulas in L(PA):
>
> pGC <-> "There are infinitely many examples of Goldbach Conjecture."

I don't know what this means. If it means "There are an infinite
number of even numbers that are the sum of two prime numbers" then
it's provable in in PA -- and true.

> cGC <-> "There are infinitely many counter examples of Goldbach Conjecture."
>
> Then:
>
> (1) <-> pGC xor cGC

--
hz
From: herbzet on 3 Jul 2010 12:45


Nam Nguyen wrote:
> herbzet wrote:
> > Nam Nguyen wrote:
> >> herbzet wrote:
> >>> Nam Nguyen wrote:
> >>>> herbzet wrote:
>
> >>>>> The whole thing is nonsense, anyway. Clearly, PA
> >>>>> is consistent, or at least, its consistency is
> >>>>> at least as evident as the consistency of any
> >>>>> system which purports to prove it.
>
> >>>> Sometimes it's much ... much simpler and more logical, humble,
> >>>> humanistic to admit we don't know what we can't know, rather
> >>>> than pretending to possess some sort of an immortal knowledge.
> >>>>
> >>>> Suppose someone states "There are infinitely many universes
> >>>> and each has harbored a planet with intelligent life in its history."
> >>>>
> >>>> If there actually are infinitely many universes we can't know such
> >>>> fact. PA's consistency is like such a statement: if it's consistent,
> >>>> you can't never know that. Period.
> >>> I see a model for PA: the natural numbers. I conclude PA is consistent.
> >>>
> >>> I would say I know this to a mathematical certainty.
> >>
> >> IOW, mathematical reasoning is just a gambling that never ends!
> >
> > You find something doubtful in the proof of the infinitude of primes?
>
> There are a lot of things I never find doubtful, such as the proof
> Ax[~(Sx=0)] in PA. But there are also a lot of theories in which not
> only Ax[~(Sx=0)] is provable but also Axy[x*y=0] is. By the natural
> numbers, would you mean they collectively be a model of such theory?

Do you mean, do I think the natural numbers are a model of the theory
which has as axioms only the two formulas Ax[~Sx=0] and Axy[x*y=0]?

> Just _exactly_ what do you "see" as the natural numbers?

0, 1, 2, 3, ...

> If there's
> a least one formula you couldn't [see?] its truth in the naturals,
> in what sense do you "know" the naturals?

I don't know my mother's blood type -- does that mean I don't know
my own mother?

--
hz
From: Nam Nguyen on 3 Jul 2010 14:07
herbzet wrote:
>
> Nam Nguyen wrote:
>> Nam Nguyen wrote:
>>> herbzet wrote:
>>>> Marshall wrote:
>>>>> herbzet wrote:
>>>>>> What is PA + (1)?
>>>>> The successor to PA.
>>>> The more I look at this, the funnier it gets.
>>> Same here.
>>>
>>>> Make it stop!
>>> Right. _Stop_ believing you "know" _exactly_ what the natural numbers be!
>> Seriously, relativity in sciences (including mathematics) isn't an
>> one-man conviction in "sci.logic", "sci.math". The mere mentioning
>> of the 5th postulate, Hilbert-era's truth-equals-provability, SR,
>> QM, should be a reminder that belief of any absoluteness in sciences
>> is an ancient belief, which is no longer adequate for describing physical
>> reality, or abstraction.
>>
>> If we scorn or laugh at the relativity of the standardness of a purported
>> "model" of L(PA), a.k.a collectively as "the natural numbers", then we're
>> no better that those who laughed at Riemann's ideas, at SR, at QM's uncertainty.
>> At least those people had a valid excuse: they were in a different time in
>> the past. We don't have such excuse!
>>
>> Seriously, all the nasty bickering aside, think about the whole thing logically.
>> Think about the 4 reasoning Principles:
>>
>> - Principle of Consistency.
>> - Principle of Compatibility.
>> - Principle of Symmetry.
>> - Principle of Humility.
>>
>> Would you think these are nonsensical principles honestly speaking?
>
> Probably.
>
> Who said anything about "absolute knowledge"? I think you're
> tilting at a windmill.

Oh, but by FOL definition of a formal system consistency, it must be
either absolutely true or absolutely false that PA is consistent!
(Ditto for inconsistency).

Of course nobody should prevent you from saying:

"I don't know"!

See how easy when we're truthful to ourself!

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