[FOM] To Vladimir Sazonov and others doubting the unambiguityof N

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Tue Jun 24 16:28:14 EDT 2003

Again, I am sorry for the late reply. 

Aatu Koskensilta wrote:
> Vladimir Sazonov wrote:
> > Aatu Koskensilta wrote:
> >
> >>Vladimir Sazonov wrote:
> >>
> >>>What (as you say) "disinformed" me? Some deeper that in the school
> >>>things like Goedel's theorems, especially on incompleteness and
> >>>Goedel/Cohen proof on independence of CH, what demonstrated (to me)
> >>>that both N and continuum are vague concepts, [ --- ]
> >>
> >>Like [] I can understand your position with regards to the continuum,
> >>but as to N, I'm still baffled. Surely the notion of *non*-standard
> >>model of arithmetic is much more illusive, as any such model must
> >>necessarily be non-recursive?

Feasible numbers 0, 0', 0'', AND SO ON, 
are a nonstandard model of (of course, not Peano) arithmetic. 
But they are quite real, not illusive. 

> >
> >
> > Here I feel you assume ZFC or the like where what you mentioned
> > makes sense. As I already wrote (actually many times), in this
> > framework I have no problems with understanding the concept of
> > standard or nonstandard models of PA. All of this is defined
> > (or proved) in ZFC quite precisely. 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).
> This is exactly what I fail to comprehend. It seems that since we have
> the metatheorem in systems like ZFC and the like about PA that all its
> recursive models are isomorphic and *standard*, this shows that we
> *can't*, without assuming quite strong platonistic framework even
> produce any non-standard models for display, or to have any point of
> ambiguity. 

First, I cannot understand which way something proved in ZFC, even 
about standard model of PA (quite a legal, INTERNAL object of ZFC), 
is related with what you are asking. This seemingly means that you 
assume some ABSOLUTE, EXTERNAL to ZFC standard N. 

On the other hand, for BA = Bounded Arithmetic which I mentioned in
another posting, take BA + ~EXP. Forget about existence of ZFC with 
all its model theory, and imagine a "model" of this theory with
exponential *partial* recursive function. Quite reasonable, intuitive 
theory which even better corresponds to our real world. This 
imaginary model is quite different from any imaginary model of PA 
and can be called, by this reason, non-standard. No strong 
platonistic framework is assumed. 

By the way, it is an open (quite precise) question whether each 
finite binary string is computable by a the Universal Turing 
machine with unary strings as inputs in the framework of this 
theory (assuming that BA has some nontrivial version of bounded 

You can do the same with PA and some provably (in a bit stronger 
theory than PA) total recursive function which is not provably 
total in PA (not epsilon-zero recursive). This will give a 
nonstandard (imaginary) "model" of PA. Of course, PA + ~Consis(PA) 
may be considered, too. 

But the main problem consists in asserting some absolutely 
unnecessary super beliefs which are coming rather from philosophy 
of mathematics rather than from mathematics itself. 

Let us recall, whether non-Euclidean Geometry "manifested itself" 
before it was discovered? Just vice versa, there were philosophical 
views just against this. Just Euclidean Geometry was considered as 
the only possible in principle, some ABSOLUTE. This was a serious 
problem for Gauss to publish his results. Lobachevsky, who dared 
to publish his Imaginary Geometry was considered crazy. 

Recall also the absolute space-time. If the eyes of scientists 
would be opened for potential possibilities, the Relativity Theory 
could arise earlier, first as an imaginary possibility. Just 
ask the question what does it mean the absolute space-time. 
which exactly experiments confirm that there is the unique time 
for distant physical objects. After asking such "stupid" questions 
and finding appropriate experiments we immediately come to the 
alternative possibility, just in principle because the experiments 
could, in principle, give different results. 

Not asking such questions means relying on some beliefs, like 
the existence of a God who is the carrier of the absolute time. 
Which other way it could be absolute. Only signals with an 
infinite speed could be the reason for the absolute time. 
But the infinite speed is something suspicious. Equally 
suspicious for me are absolute truth or absolute, standard N. 
Should we learn something from the history of Science and, 
in our case, from what was happened with CH? 

When you work in ZFC, you seemingly consider just some imaginary 
universe of sets. Which serious reason (besides any your beliefs) 
does not allow you to think the same way on N and PA? 

I don't think we need to relativise these notions to any
> specific metatheory, even though the result itself is proved in some
> metatheory.
> You think that N is in some sense vague. How does this vagueness
> manifest itself? Are there some theorems of arithmetic that don't have a
> specific truth value? Are there some identities that aren't "decided"? I
> can't see how N can be vague *unless* one assumes quite strong a theory,
> in which it's supposed to be so (say ZFC).

Of course, if we would find an undecidable hypothesis (may be P=/=NP 
is such one) for which it will be demonstrated, as for CH in ZFC, 
its independence of PA and even of ZFC and of many reasonable 
extensions of ZFC, then we will have some demonstration of the 
vagueness of PA which would convince many people. Is such a possibility 
absolutely excluded? What is PA is in fact inconsistent? What then 
would happen with this "standard" N you believe in? 

I believe that any super beliefs are harmful. There is no rational 
reasons for them. We will loose nothing if we reject these beliefs. 

>  > 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.
> > However, I seemingly already explained my views in all
> > essential details. I am just wondering why what I wrote
> > quite explicitly is misunderstood or just ignored.
> I don't think I misunderstand you as thoroughly as you think. In my
> opinion the fact that all standard models of arithmetic are recursive is
> of philosophical importance, especially for someone with a
> finitistic/intuitionistic/constructionistic stance. You might of course
> disagree, but I don't see why. Supposedly recursiveness is an absolute
> epistemological notion. Do you agree with this? 

No! This is one of many mathematical notions which were formalized 
successfully. This notion is extremely important. But there is no 
reason to use the word "absolute". 

Or do you think that
> recursiveness itself is in some sense relative to a formal theory?

Yes. I demonstrated above how this is possible. 
Anyway, this is one of mathematical notions and 
can be considered ONLY in a framework of a formal theory. 
Otherwise, what could you ever say about this notion 
outside a formal (rigorous) reasoning? Trying to do that 
is only self-deception. 

> What you write is misunderstood or just ignored because it's alien, at
> least to me (insert smiley here). 

What is so alien? If you understand the universe of ZFC as something 
not absolute, what is the problem to you to understand N in the same 

At first I thought you represented
> some sort of Hilbertian formalsim (an eminently sensible view, by the
> way), but your views now seem much more radical.

It seems the only difference is that I pay attention to the fact 
that derivations in a formal system are of feasible length. 
Although this looks a trivial note, other people have a problem 
with this because of knowledge of metamathematics where derivations 
can be just imaginary ones. They identify imaginary with real 
(what actually was noticed by Dana Scott as infinite regress). 
Additionally, many want to have something absolute, if not the 
universe of sets, then, at least, N. But why it is considered 
so important??? 

> >>There seems to be no such clear distinction with standard and
> >>non-standard models of set theory (let alone the notion of "the"
> >>standard model of set theory), and thus I can appreciate the idea that
> >>there is something inherently vague to the continuum or the even more
> >>substantially infinitistic set theoretic objects. But N? There seems to
> >>be a genuine *mathematical* distinction here; the standard model is the
> >>recursive model, and the non-standard ones are the non-recursive ones.

I could only repeat that this is an internal fact of ZFC. 
Externally, N considered in ZFC is just an imaginary object. 
Externally, we have no rational way even to formulate what 
does it mean that N is standard one, and there is nothing 
to do as to consider N as imaginary one. How to compare 
imaginations of various people whether they are the same? 
We can compare only formal systems which we are using. 

> > The fact that WITHIN ZFC we have these results has no relation to
> > my question on what is the ABSOLUTE standard N. I strongly believe
> > that this is actually a wrong, fictitious concept having nothing
> > rational behind of it. Usually mentioned abstractions of potential
> > or actual infinity with respect to the ABSOLUTE standard N are
> > themselves very vague.
> But the absolute standard N only becomes vague when you wander outside
> the realm of intuitionistic/constructionistic/finitistic mathematics!

WHY??? How is it ever possible to define what is standard N in the 
finitistic theory PRA? This is possible only in ZFC or the like!
What did you mean at all in the above phrase? 

> This is very much unlike the case with ZFC and its standard and
> non-standard models, let alone its "the standard model". You seem to see
> no essential difference here, which baffles me.

I do not understand which difference (which I "do not understand") 
do you mean. 

> > Say, potential infinity of N essentially assumes that we can
> > always add 1 to any number. Moreover, it is assumed that we can
> > ARBITRARILY iterate this our ability. It is this ARBITRARILY what
> > is unclear for me. I understand that this assumes that by iterating
> > the operation x+1 we should have the ability to always fulfill the
> > operation x+y. Further iterations lead to multiplication,
> > exponential, superexponential operations, primitive recursive
> > functions, Ackermann's function,..., epsilon-0-recursive functions,
> >
> >                       AND SO ON.
> >
> > 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 don't see why this needs to be vague at all. It is true that many
> philosophers in the past have formulated this "going on arbitrarily" in
> a misinformed fashion, even claiming that we are somehow "compulsed"
> mentally to always go on. This compulsion exists, but it is of modal
> nature; we *can* always go on, and this continuation is unambiguous
> *provided* one doesn't actually work in some strong platonistic
> metatheory. 

Just vice versa, this may be done unambiguous only when we 
formalize (explicate) this process in some way. This may be done 
only in (or relative to) a sufficiently strong (meta)theory. 

The so-on simply means that we *can* always consider the
> natural numbers further ahead, and that they are uniquely determined by
> the production rule (and the fact that whatever recursive function we
> might consider, it acts "correct" for these numbers, i.e. in effect
> Dedekind's recursion theorem).

How can you discuss all of this outside a formalism? 
I also tried to discuss above the potential infinity, 
outside a formalism, but I honestly said that I am unable 
to do this in a definite way. I do not know what does it mean 
this "and so on" in general. You seems assert that you know,  
but by which miracle? 

> > 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.
> I can't understand how any number we can actually name and work with
> could be non-feasible!

Is this the question on the (very informal) definition I already 
presented and explained in detail, or only on the terminology? 
Will you be precise, please. 

I also wrote that natural numbers, like this one, are quite 
legal objects of PA. Why should I repeat? 

> > Of course, we can, in principle, make this AND SO ON explicitly
> > defined WITHIN a formal theory. But this will mean that we
> > relativized this AND SO ON and corresponding version of N to a
> > formal theory. (QED!) In general, the only possibility to do
> > something precise in mathematics is via formalizing.
> Your conclusion only follows if one accepts your thesis that the so-on
> is somehow inherently vague. I don't see why this needs to be so.

This is probably because you are too religious concerning N. 
Any rational questions concerning the "holy" N are forbidden? 

> >> From your postings I gather this won't satisfy you, but I'd be
> >>interested to know whether Gödel's theorems merely motivated you to
> >>question the platonistic picture of mathematics or do you believe they
> >>server as arguments against such a position?
> >
> >
> > As I wrote in a posting to FOM, they are (may be indirect) witnesses
> > of the vagueness of N. They stimulated me to start doubting and asking
> > the question "what is the standard model of PA?". I have no direct
> > answer, and, I believe, nobody has. Why then to use this "wrong"
> > concept (except explicitly within ZFC) at all?
> There is a kernel of truth in what you say. I believe we don't need a
> concept of "the standard model of PA" unless we're already working in a
> strong theory, 

looks like you understood me, but...

>simply because there's no way for non standard models to
> arise.

No, simply because there is no rational way to explain what it is, 
like there was no rational way to explain why the only possibility 
for the geometry is to be Euclidean. 

You seems want to have any belief, if there is nothing (from your 
point of view) against it. 

I, as soon as I able to, avoid any beliefs in mathematics and 
science. There are quite rational theories which replace them. 

Down with irrational super beliefs in Mathematics! 
They are not only useless and meaningless, but potentially harmful, 
like the beliefs in the unique geometry or in absolute space time. 
This view is much more healthy. 

> --
> Aatu Koskensilta (aatu.koskensilta at xortec.fi)
> "Wovon man nicht sprechen kann, daruber muss man schweigen"
>   - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Vladimir Sazonov

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