[FOM] Query for Martin Davis. was:truth and consistency

[Vladimir Sazonov]

Dear Robin Adams, 

You posting looks like taking my "rules of the game" to demonstrate 
that this leads to some absurd conclusions. But you wrongly 
understood my rules. I think that I already answered to you 
posting in general in my reply to Bill Taylor. I do not know, 
whether this is enough for you. Give me know, if it is not the case. 
Here I will give only some comments related to your posting. 

Anyway, if you want to understand what I mean, I think you 
should try to "play this game". 


> Suppose I propose the following formalism.  Define the formal system N as
> follows.
> 
> The set of terms is defined thus:
> 
> 1) 0 is a term.
> 2) If t is a term, then t+1 is a term.

Yes, this could be considered as a formal system consisting 
of two formal rules of "inference". Here inference is, of course,  
not of something which is intuitively "true" or "false". You 
infer just numerals (your terms). 

> 
> and the usual rubric about the set of terms being the smallest set closed
> under these two clauses.

If we are working not in a metatheory of this formal system, then 
this rubric is unclear. Your even use the term "set". Therefore 
you assume a metatheory (on the formal system N) involving this 
concept, i.e. some set theory, or some second-order theory. But 
we can understand what any particular formal system is without 
any metatheory.  

I consider a formal system as a finite object (in your case, the 
above two lines), actually a generation rule. We can (physically) 
generate in this system only feasible terms (numerals). 
We will never be able to generate a numeral intuitively 
corresponding to 10^10^10. When I refer here to 10^10^10 
I mean that in some formal systems like PA we know the 
definition of this abstract object (of PA). Your numerals (terms) 
correspond to some other abstract objects of PA and in PA 
we can evidently prove for EACH derived numeral that the 
object it denotes in PA (i.e. a natural number) is < 10^10^10. 

I hope, this is clear. If you do not agree with this conclusion 
(from my premises), then you shold present a counterexample: 
a (physically written) numeral and a proof in PA that it 
is >= 10^10^10. 


<some stuff omitted>


> I'd like to end with a specific challenge.  You seem to accept the reality
> of PA, but you have said that there is a sense in which you do not accept
> the reality of 10^10^10^10.  Consider the following formula:
> 
> x_1 = x_1 & x_2 = x_2 & ... & x_(10^10^10^10) = x_(10^10^10^10)
> 
> where x_1, x_2, ..., x_(10^10^10^10) are distinct variables.  I claim that
> this is a formula of PA, and is provable in PA.  Do you agree.  


I would agree if you would write this "formula" explicitly 
without "...". Also I would immediately agree if to consider 
this in an appropriate metatheory. 

If so,
> what is its length?  And what is the length of its proof?

I could consider its length and the length of the proof only in 
a metatheory. Otherwise, it is meaningless to discuss the length 
of the above "formula" 


> 
> If you cannot answer these questions, how does this square with your
> belief in the reality of PA?  

All of this was answered (PA, as well as your N, is a finite, 
even feasible object - a system of formal rules),  

If you can, how do you reconcile this with
> your problems with the existence of 10^10^10^10.

and this, actually, too. The number you mean by this expression 
has NO counterpart (as a numeral) in the real world. (Of course, 
I do not mean here the syntactical expression 10^10^10^10 which 
does really exist.) But in the framework of a formal theory like PA 
we can define the corresponding number. By the intuition related 
with PA we can also imagine it as "existing". But this imaginary 
"existence" is of a different character than existence of 
particular (feasible) numerals of your formal system N. 

I hope you already understood me. 

Of course, you may disagree with my rules of the game. 
I only want to say that these rules are sufficiently clear 
consistent and reasonable, and that I see no better alternative, 
even no consistent alternative - without introducing meaningless 
fictions like "the standard model" of PA (in some absolute sense). 


> --
> Robin Adams


Vladimir Sazonov