[FOM] Godel Sentences

[Kanovei]

[FOM] Godel Sentences

It seems to be an appropriate time to come back to the claim 

(*) there exist true, but unprovable sentences of PA 

which, surprisingly, persists, in spite of clear evidence of otherwise, 
in the postings even of undoubtfully educated specialists in foundations. 

A.P. Hazen gives the following example:

>>>>>>>
        "This reminds me of the mother who, trying to
        get a balky child to eat its dinner, says "Think of
        all the starving children in Ethiopia who would
        just LOVE to have a bowl of spinach as nice as that
        one" to which the  brat replies "Name one.""
Starving children tend not to be known by name to residents of developed
countries, and it is of course contradictory to suppose that one could
DEMONSTRATE that a proposition is both true and  unprovable.
>>>>>>>>>>

What on Earth prewents the hapless mother (together with Prof. Hazen) 
to take a jet to Ethiopia, then a cab up to a nearby willage, where 
the mother (if not Prof. Hazen) will know as many names of children 
starving, dying, or dead already as they need to get a young 
*resident of developed countries* back to his spinach.  

Or Prof. Hazen assumes that the *residents of developed countries* 
now needn't take time and trouble to getting knowledge they are 
looking for ? 

Perhaps opposite to what Hazen expected to support by this exaple, 
in fact the example demonstrates the crucial difference between the 
problem of ethiopean names (which can be easily solved in the positive 
by an appropriate example obtained as above), and the problem of (*) 
which answers in the negative because there is NO 
true, but unprovable sentences of PA .

>>>>>>>>>>
So I do  not accept the challenge 
[The challenge to present an example to support (*) - V.K.]
>>>>>>>>>>>>>

You quit the challenge simply because you do not have an example 
AND MOREOVER do not see any clear way how such an example can be 
obtained.

>>>>>>>>>
Ironically, G=F6del there sides with Kanovei ...
>>>>>>>>>

There is nothing ironical that those who understand the problem 
sideline opposite those who does not seem to.
(The difference in rank in the scientific hierarchy is recognized 
and not disputed.)  

-----------

H.Friedman:

>>>>>>>>>>>
Kanovei confuses two issues. There is the well known ROBUSTLY FORMULATED
theorem, call it T, of mathematics, fully accepted by the mathematics
community, that there exist true sentences in the language of PA, not
provable in PA. But then there is the issue of presenting an example that
meets certain criteria.
>>>>>>>>>>>>>

I have never mentioned the last issue, which is irrelevant to the 
problem at all (although important in different other aspects). 
Yet the wrong claim of Friedman contains a sort of clue to the problem 
(for those who follows clues). 
Yes, there is a theorem (Goedel's theorem) that 
there exist true sentences in the language of PA, not provable in PA, 
or, technically, 

(**)  \exists \phi 
      (\phi is a closed PA formula such that 
       1) N\models \phi and \neg Prov(PA,\phi))

where \models and Prov (and everithing else) are well defined notions, 
say, in ZFC. 

Yet why this (undisputable) theorem implies (*) above ? 

To explain the point, I recall another (also undisputably established) 
theorem, that there exists a Banach-Tarski decomposition of a sphere. 
It would be highest degree of ridicule to claim, on the pretext of 
this result, that a real plastic ball can be decomposed this way. 
(Although I came across a reading which tried to explain some 
biblical fabules this way.) 

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

Arnon Avron: 

>>>>>>>>>>>>>
Kanovei writes:

> Those who believe  in the paradigm
> *there exist true, but unprovable sentences of PA*
> in its straightforward sense
> are welcome to kindly present such a remarkable sentence
> along with a demonstration of its desired properties.

I just could not believe my eyes when I read this. Are you serious???

Well, some version of Consis_PA is a sentence like this
>>>>>>>>>>>>>>>>>

Any version of Consis_PA is true only because it is provable in 
a higher system (say in ZFC), or because it is just matter of 
pure belief (for those who do not like to employ ZFC here). 
I do not see any point in going the latter way because it is 
absolutely to the contrary of rules of mathematical community 
to accept that anything is true because of widespread belief only. 

>>>>>>>>>>>
Now the only reasonable notion of "truth of sentences of PA" is truth
in the structure of the Natural numbers (the "standard model"). And
I simply cannot see that one can accept the Natural numbers and still
honestly doubt the consistency of PA. So if arithmetical truth has 
any meaning at all, then PA is consistent, and from this one can infer
that Consis_PA is true (the last step of showing that if 
PA is consistent then Consis_PA is a true arithmetical sentence
>>>>>>>>>>>>>>>

This is a rather awkward outline of the proof of Consis PA in ZFC 
(or in a weaker, but sufficiently strong subtheory), 
which is a well known fact. 
Does it help in any way to give an example for (*) ? 

V.Kanovei