[FOM] To Vladimir Sazonov and others doubting the unambiguity ofN

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Mon Jun 23 17:00:59 EDT 2003

Sorry, by some reasons I reply rather late. 

Lucas Wiman wrote:
> On Sunday, June 15, 2003, at 10:40 AM, Vladimir Sazonov wrote:
> > Aatu Koskensilta wrote:
> >> Surely the notion of *non*-standard
> >> model of arithmetic is much more illusive, as any such model must
> >> necessarily be non-recursive?
> >
> > I have a serious problem
> > with understanding when "standard" model of PA is mentioned
> > in some ABSOLUTE, metaphysical, quasireligious sense (not
> > RELATIVE to ZFC or the like). It seems you do not understand
> > what I mean. If I would know what is unclear for you when I
> > refer to ABSOLUTE and RELATIVE, I would try to explain.
> Recursiveness is supposed to be an epistemological notion, though one of
> great mathematical significance.  Interestingly, it also seems to lead
> to nontrivial metaphysics in this case, since "the standard model of PA"
> becomes "the only one we can construct."  No religion.  Nothing like
> that.

Sorry, bur I see here only a declaration on a construction of 
"the standard model of PA", no construction at all. 

> It seems you must have a serious counterargument to Church's thesis or
> Turing's thesis, which would be a significant event for philosophy
> generally.

Church's thesis, as I understand it, is nothing else as the 
statement of a high satisfaction by the formal definition 
of computability given either in terms of lambda calculus 
or in terms of Turing machines or the like. We have analogous 
situation  with the definition of continuous functions. We just 
assert that the definition corresponds to our intuition well 
(however the initial intuition may be somewhat changed; before 
giving these definitions it was not assumed that computable 
functions are inevitably partial and that there should exist 
continuous, nowhere differentiable functions, etc.). 

By the way, in Bounded Arithmetic we also can define computability 
by Turing machines. But exponential 2^x is not provably recursive 
here. It is computable, but not provably total. Therefore it is 
possible and quite reasonable in this framework to postulate it 
to be partial (this is written as ~EXP). Strictly speaking 
Church's Thesis says nothing about totality of this function. 
Moreover, we have universal Turing Machine here satisfying 
s-n-m theorem and recursion theorem. However, because of partiality 
of exponential function, we should be little bit more careful 
with the definitions. 

Thus, CT is a relative thesis. It is meaningless to consider 
it in an ABSOLUTE way. At least, I know no explanation what 
it is absolutely.  

> > Yes, we can continue further and further, but how further?
> > Until we will get tired? What this AND SO ON really means?
> > Can anybody explain? If not, then this is something indefinite,
> > vague. Thus, the "resulting" N is also vague. Let us be honest
> > before ourselves.
> I agree with Bill, you're definitely an ultra-finitist.

Is it your reaction on my question "What this AND SO ON really 
means?" Have you any answer instead of branding? 

> > Again, what is the "length" of the resulting N? Intuitively,
> > it is much more comfortable for me to think (together with
> > Esenin-Volpin) about many (infinite) Ns of various "length",
> > with various abilities to iterate the ability to iterate the
> > operation x+1. It is intuitively plausible that the simple
> > iteration of x+1 leads us only to feasible numbers where
> > 2^1000 is non-feasible.
> What?  A number you just named is non-feasible?  This makes absolutely
> no sense.  I ask that you give me a definition of non-feasibility.  

It seems I already wrote enough about this in my recent postings. 
Anyway, the numerals 0, 0', 0'', etc., which we (or a computer) 
are able to write explicitly, physically are denoting feasible 
numbers. Other numbers are called non-feasible. Of course, this 
is a vague concept. Nevertheless, it is sufficiently clear because 
we refer to something concrete. Now, will you write, please, a 
numeral denoting the number which is also denoted (in PA or ZF, 
etc.) by 2^1000 or even 2^100. 

> any definition you give me, I can name a larger number than that, and it
> is in that sense I think the idea of an "absolutely non-feasible number"
> makes no sense.

I defined above nothing absolute. I repeat, that the concept 
of feasible number is vague. But it is not so vague to assert 
that nobody will be able to write the numeral denoting 2^1000. 
This is like a law of physics. 

> Let's think about the number 2^127-1.  This number was proved prime by
> Edouard Lucas in the late 19th century, by hand.  However, he used a
> method which did not check for all prime factors, in fact using
> elaborate calculations on a a group of size 2^127 (actually his proof
> was to show that this group does indeed have this size).  You seem to be
> saying that none of these results makes sense because he couldn't
> possibly write down a string of 2^127-1 marks.  

Aha! You actually understand what is non-feasible finite object. 
Then, why did you ask me above? 

> Or am I misunderstanding
> you?

Did you read my last posting to FOM with enough attention? 
How is it possible to misrepresent my point of view so badly? 
I wrote that, as a formalist, I accept ANY formal system which 
formalizes some intuition, in partular PA. The above result 
can be formulated and, most probably, proved in PA. I have 
enough of intuition on PA (without any reference to "standard 
model" of PA; only some idea on an imaginary universe for PA) 
to understand the meaning of this result and your comments how 
it was proved. 

> But I can tell you things about vastly larger numbers.  I can tell you,
> for example (without doing the calculation!) that 3^(2^127-2) has 1 as
> the remainder when divided by 2^127-1.  This is all basic number theory,
> yet it easily transcends the epistemological limitations you put on
> mathematics.  You're denying mathematical facts, 


and hence your
> epistemology is false.  There are certainly limits to my ability to name
> numbers (for example, there are numbers on the order of 3^2^127 which I
> cannot name), but there are gaps and in my ability to name them, I can
> still name much larger ones.  If someone could give this a precise
> formulation, it would be interesting, but it would not have the
> epistemological weight that you give it.
> You'll (no doubt) say that I'm "really" working in PA above, though
> this, too, denies mathematical reality.  (I know, for example that
> addition is commutative and associative, but I don't know the proof of
> it in PA.)  

I already discussed relations between formal and semi-formal 
reasoning in mathematics in my postings and do not want to repeat. 

As to mathematical reality, it is nothing else as our formalisms 
+ intuitions about them. My mathematical reality is RELATIVIZED 
to formalisms. Nothing essential is denied. The ABSOLUTE reality 
of a mathematical world is denied. Bit nobody can give it a 
clear meaning. It is just a kind of unnecessary mathematical 

If you explain the proof of Godel's first incompleteness
> theorem to most mathematicians, they'll say something like "yes, but
> then the Godel sentence has to be true."  What this tells you is not
> that mathematicians are working in second order number theory (because
> one can obtain a Godel sentence about that, and a mathematician will
> tell you the same thing), but that mathematicians aren't working in a
> formalized system at all.  If one thinks of logic as the "projection" of
> mathematical arguments, it's interesting to study, but it is objectively
> different from real mathematical arguments.  Gauss never needed PA to
> understand arithmetic. 

Did Gauss present non-rigorous, non-formalizable proofs? 
Why should I repeat these trivial things? 

> It's interesting to note that Vladimir has had precisely the opposite
> reaction of some notable mathematicians to Godel's theorems.  Alain
> Connes (a Fields medalist), for example, said in a recent book that
> because of the ability to see that Godel's statement is true,
> mathematicians must in fact be accessing some kind of reality separate
> from formal systems.  No formal system can totally capture all the
> ability of mathematicians have to reason about mathematics.

The main idea of formalism (at least, how I understand it) 
is based also on realizing that there could be no unique 
formal system for the whole mathematics. However, numerous  
formalisms can cover mathematics which is based on rigorous 
reasoning, that is, ideally, on this or that formalism. 

As to accepting Goedel sentence or Consis(PA) to be true, i.e., 
to be a new axiom, it is an interesting phenomenon 
by which we extend a formalism (here - PA). It is related with 
a strong extrapolation. Both these (universal) sentences are 
about formal proofs in PA, but, strictly speaking - also about 
nonfeasible, imaginary proofs which are NOT genuine proofs. 
By this reason I would not read Consis(PA) literally as "PA is 
consistent". Nevertheless, psychologically, there is a strong 
temptation to accept much more strong sentence Consis(PA) as 
a new axiom. There may be some doubts as well. 

Of course, this step of extending PA is not a rigorous 
mathematical step. But this is a step of our intuition. 
We always use our intuiton when creatinf a new or extending 
an existent formal system. 

This is only little bit more complicated than the ordinary 
phrasing of the situation. But this is not so difficult 
and, I believe, is more reasonable and productive than 
referring to fictions like the "standard N" and truth 
of Consis(PA) in it. As I explained above, I am not so sure 
that Consis(PA) is true in my imaginary "model" N of PA. 

On the other hand, the traditional phrasing on the truth of 
Consis(PA) is also possible, if we take ZFC as a metatheory 
of PA and define, as usually, the standard model of PA and 
the the model-theoretic concept of truth. Then this will be 
just RELATIVISED to ZFC considerations. (This is so trivial, 
that I feel myself very uncomfortable to repeat this.) 

What is so difficult and not understandable here? 

> - Lucas Wiman

Vladimir Sazonov

More information about the FOM mailing list