[FOM] First-order arithmetical truth
V.Sazonov@csc.liv.ac.uk
V.Sazonov at csc.liv.ac.uk
Fri Oct 20 14:19:20 EDT 2006
Quoting "Timothy Y. Chow" <tchow at alum.mit.edu> Thu, 19 Oct 2006:
>>> If you lack the ability to distinguish the intended model of PA from
>>> another model,
>>
>> Excuse me Timothy, please. Do YOU have this ability?
>
> For the purposes of the point I'm making here, let's assume that I *don't*
> have this ability.
That is, you postpone answering my question. Or is this an attempt to
lead me to a contradiction?
That's what I need to make my point. From this
> inability of mine, I infer that I am also unable to distinguish the
> intended meaning of "formal system" from an unintended meaning of "formal
> system." Do you disagree with this inference?
I agree (if to play in this game).
But we should make a clear distinction between (meta)mathematical
concept of a formal system (say, a specific version of predicate
calculus) and the corresponding naive concept. The former concept is
quite similar to that of mathematical natural numbers, and, like for
numbers there is no possibility to distinguish between standard and
non-standard. (More precisely, the concept of "standard" natural
numbers or formal system makes no sense at all if considered as an
absolute concept).
On the other hand, the "naive" concept of formal system (considered not
from a metamathematical perspective) is highly important to explicate
what is mathematical rigour. We understand any formal system by
considering (or learning) what is the underlying formal language and
proof rules just by examples, by training. This is a kind of real
physical manipulation with some specific finite objects very much like
children or all of us can play with lego, domino, card games, chess and
understand the general idea of such kind of activity without going into
any theoretical and mathematical considerations. Of course, any formal
proof has in this case only a feasible length. What means "feasible" is
rather vague idea. But what is a well-formed formal proof is
practically well understood and highly reliable ground for presenting
rigorous mathematical proofs despite all of this understanding is quite
naive.
Evidently, metamathematical concept of formal system (which is also not
unique because depending on the underlying theory where it is
formalised and its properties, such us cut elimination, etc. are
proved) is very different from this naive one. For example, in the
naive formal logical calculus cut elimination is practically
impossible. When we assert that a "naive" formal system is inconsistent
we assume that a concrete physically presented formal derivation of
inconsistency can be presented. Consistency assumes a naive universal
quantification over all feasible derivations. All of this is quite
understandable, however rather naive both because everything is based
on examples and training only and because of feasibility implicit.
Or do you claim that you
> have some sort of mystical ability to distinguish "standard" formal
> systems from "nonstandard" formal systems?
I do not know what did you expect from me by asking this and by your
inference above.
I have no mystical ability at all. I honestly assert that my
understanding of the fundamental concepts of mathematics is based on my
intuition and imagination which are rather restricted. What I imagine
when think on natural numbers is something vague especially when going
to very big numbers or to infinity. Nothing "crystal clear" or
"standard" (in infinity or for large numbers). On the other hand I have
quite solid and reliable ability to recognise a rigorous mathematical
proof (as anybody here; nothing mystical!). I have repeated in FOM
infinitely many times that I absolutely do not understand what other
call the "standard model" (as something absolute, ideal and uniquely
existing). I see that nobody is able to explain what it is and see no
scientific ground allowing to do that. I believe that this is
absolutely meaningless and mystical concept. I am puzzled why people (I
know that not all of them!) concerned with f.o.m. are so inclined to
mysticism.
On the other hand, I would rather apply the term "standard" to the
above *naive* concept of formal system (and to analogous *naive*
concept of feasible number) because it is based on a quite concrete and
well understood (let even naively understood) human practice and common
sense. This naive concept of formal system is the real and sufficiently
solid ground for understanding more explicitly what is mathematical
rigour and, I believe, this should be the starting point for an
adequate philosophy of mathematics.
But when discussing on formal systems, let us not mix mathematics
(based on the naive formal systems) with metamathematics (considering
formal systems as mathematical objects).
Finally, it is quite easy and well-known how to present conservative
extension of PA whose models (let understood informally) are
non-standard (so to speak, "somewhat strange"). In this sense existence
of non-standard models of PA is easily "observable". Nothing analogous
can be done for "standard" and, I repeat, it is even absolutely unclear
what does this "standard" ever mean.
Moreover, feasible numbers are in fact the initial part of ANY
(imaginary) model of PA and even can be consistently postulated (or
just imagined) to be both upper bounded and closed under successor. In
this sense ANY model of PA behaves as non-standard. Nothing mystical,
except that some formalisation should be presented to make a support
for imagination.
Vladimir Sazonov
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