FOM: Proper Names and the Diagonal Proof
Insall
montez at rollanet.org
Wed Jun 26 16:00:49 EDT 2002
Allen Hazen has done a nice job of trying to explain, in an informal manner,
a reasonable interpretation of Cantor's diagonal argument for fairly strict
anti-Platonists. A more formal explanation follows from a simple
cataloguing (below) of various well-known results, or various minor
modifications thereof. To read this, observe that we use the following
abbreviations:
CT ~ ``Class-Set Theory (without Fraenkel's Axiom of Substitution)''
(as described in Goedel's proof of
the relative consistency of the GCH)
CTF ~ ``Class Theory with (the class version of) Fraenkel's Axiom of
Substitution''
CTFC ~ ``'CTF with the Axiom of Choice'
Z ~ ``Zermelo Set Theory''
ZF ~ ``Zermelo-Fraenkel Set Theory''
ZFC ~ ``Zermelo-Fraenkel Set Theory with the Axiom of Choice''
Z+(V=L) ~ ``Zermelo Set Theory with the Axiom of Constructibility''
ZF+(V=L) ~ ``Zermelo-Fraenkel Set Theory with the Axiom of
Constructibility''
Results proved long ago:
1. CT is a conservative extension of Z
2. CT is finitely axiomatizable.
3. Z is not finitely axiomatizable.
4. CTF is a conservative extension of ZF.
5. CTF is finitely axiomatizable.
6. CTFC is a conservative extension of ZFC.
7. ZF is not finitely axiomatizable.
8. ZFC is not finitely axiomatizable.
9. It is provable in Z that if X is a set and P is its
power set, then there is an injection from X into P, but
there is no injection from P into X.
10. It is provable in CT that the set of denumerably
long binary sequences is in one-to-one correspondence
with the power set of the natural numbers.
11. The Lowenheim-Skolem Theorem is provable in CTC, ZFC, etc.
A consequence of this is that whether you are a Platonist or not, if you use
classical logic, then your ``describable'' denumerably long binary
sequences - i.e. those which are in the constructible universe - cannot be
constructibly (i.e. ``describably'') placed in a one-one correspondence with
the set of all natural numbers. This holds even if there is a bijection
between the describable denumerably long binary sequences and the natural
numbers, for in this case, the bijection in question is merely
``indescribable'' (not constructible).
Of course, Goedel's argument for the undecidability of Peano Arithmetic
basically does the same thing.
One effect of the Lowenheim-Skolem Theorem (LST) is to formalize reasons for
``paradoxes'', such as the so-called ``Skolem's Paradox'', in which a
countable model of set theory exists, but in that countable model are sets
whose denumerability is not ``witnessed'' by a function that resides inside
the model at hand. (The failure of any such model to include such a
``witnessing'' function can in fact be formalized, to show that a certain
string of symbols in some specific language is recursively generated from
some other specific string of symbols according to a specific set of
instructions. Then one may ignore the existence of the human being with
intuition about functions, and plod along mechanically, with no meaning
behind one's formalisms whatsoever. This makes the anti-Platonists happy, I
guess, while for the Platonists, I expect it is somewhat boring, though I
could be wrong.)
What I do not understand is why anti-Platonistic Formalists are asking for
``real meanings'' or ``real objects'' to go with these mathematical
concepts. Once the definition is written, it seems to me, the only
``object'' that the Formalists I have come to know and love will accept is
that definition itself - the string of marks on the page that they can sense
by some presumably physical means. Now, I do realize that there are
pseudo-Formalists also - computer scientists, for example - who are using
formalism as a way to determine how to ``communicate'' with an inanimate
computation device. But Formalists, as I understand it, derive some sort of
meta-physical conclusions from the lack of understanding a machine or other
inanimate object, like a brick, has, along with some sort of universally
democratic principle that suggests that ``All are bricks.''. But to have
the principle requires that one be not as inanimate as a brick, or computing
device, so there is, inherent in the statement of the philosophical
principle that ``no meaning exists'', the contradictory notion that ``there
is meaning to the statement that ``no meaning exists''.''.
If I have misunderstood the stance of the anti-Platonic Formalists, please
help me to clear up this confusion, by telling me a more correct meaning of
the terms ``formalist'', ``formalism'', etc. I mean this not merely for
formalism in the philosophy of mathematics, but also, in more general terms.
If one cannot accept the existence of an infinite set, how can one accept
the existence of an electron, or, even better, in concert with some of Allen
Hazen's remarks, how can one accept the existence of that electron over
there?
Matt Insall
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