Date: Apr 6, 2013 12:25 PM
Author: namducnguyen
Subject: Re: Matheology § 224

On 06/04/2013 9:03 AM, Nam Nguyen wrote:
> On 06/04/2013 8:35 AM, WM wrote:
>> On 6 Apr., 16:13, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>> On 06/04/2013 7:32 AM, Peter Percival wrote:
>>>

>>>> Nam Nguyen wrote:
>>>
>>>>> But if GC is undecidable in PA, there's no proof left in
>>>>> FOL but
>>>>> _structure theoretically verifying_ the truth value of
>>>>> GC in
>>>>> this structure.

>>>
>>>> If the Goldbach conjecture is undecidable in PA then it is true.
>>>
>>> Care to verify (prove) your claim here?

>>
>> Goldbach conjecture is false. <==> Counter example exists. <==>
>> Counter example can be found. <==> Goldbach conjecture is decidable.
>>
>> The second equivalence requires to neglect reality. But in mathematics
>> this is standard.

>
> But, to start with, how would one _structure theoretically prove_ the
> 1st equivalence:
>
> "Goldbach conjecture is false. <==> Counter example exists."
>
> ?
>
> Logically:
>
> (A _specific_ counter example exists) => (Goldbach conjecture is false).
>
> How would one _prove_ ( i.e. _structure theoretically verify_ ) the
> other-way-around?
>
> Consider:
>
> (P(0) -> Ex[P(x)]) <-> (Ex[P(x)] -> P(0))
>
> Is this a logical equivalence?


Iow, the meta statement:

"This.specific_counter_example exists"

is not of the same semantics as:

"There exists a counter example".

And this is logically true, hence uncontroversial.

--
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There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
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