[FOM] To Vladimir Sazonov and others doubting the unambiguity of N
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Sun Jun 15 11:40:21 EDT 2003
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?
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). 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.
>
> 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.
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.
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.
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.
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.
>
> From your postings I gather this won't satisfy you, but I'd be
> interested to know whether GЖdel'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?
> --
> Aatu Koskensilta (aatu.koskensilta at xortec.fi)
Vladimir Sazonov
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