>From: Bruno Marchal <marchal.domain.name.hidden>
>To: everything-list.domain.name.hidden
>Subject: Re: Why is there something rather than nothing?
>Date: Thu, 20 Nov 2003 12:57:55 +0100
>
>At 18:30 19/11/03 -0500, Jesse Mazer wrote:
>>Does anyone know, are there versions of philosophy-of-mathematics that
>>would allow no distinctions in infinities beyond countable and
>>uncountable? I know intuitionism is more restrictive about infinities than
>>traditional mathematics, but it's way *too* restrictive for my tastes, I
>>wouldn't want to throw out the law of the excluded middle.
>
>
>I don't know. In general people who accept
>the uncountable accept most big cardinals.
>This follows from the fact that in most set
>theories you can prove Cantor theorem
>which say that card(power of A) is always
>strictly bigger than card(A).
But as I said earlier, the power set of a countable set would include plenty
of sets that could not be defined using any sort of finite description, and
I'm skeptical about the "existence" of such objects in a Platonic sense.
Perhaps what I'm talking about would be a form of "mathematical
constructivism", although I don't know how "constructible" is usually
defined, and whether it would allow for noncomputable numbers that can still
be defined in terms of some finite program of an oracle-machine whose
"level" can itself be finitely defined (see my post at
http://www.mail-archive.com/everything-list.domain.name.hidden/msg04809.html for
what I mean by different 'levels' of oracle-machines, I'm not sure if my
terminology is correct here although I've seen similar ideas expressed
before).
On second thought though, if I am only willing to admit mathematical objects
with finite descriptions, then I should probably not even distinguish
between countable and uncountable sets, since the collection of all possible
finite descriptions must be countable. But there is another sense in which
it makes sense to distinguish two different types of infinity here. To prove
that a set is countable, one must find a function mapping the positive
integers to the members of the set, with every member corresponding to one
integer. But if only mathematical objects with finite descriptions are
permitted, this should go for functions mapping integers to mathematical
objects as well; and it's relatively easy to see that no
finitely-describable function would include *every* object with a finite
description (but no meaningless descriptions which don't pick out
mathematical objects at all), using a diagonal argument. Suppose we just
want to come up with an exhaustive list of all the finitely-describable
reals, and we have some candidate function which maps every positive integer
to a real with a well-defined finite description ('well-defined' meaning
that there is no ambiguity about what number the description picks out--'a
number bigger than 4' would not be a well-defined description, for example,
since it doesn't pick out a unique real). If the function F is itself
clearly defined, so there is no ambiguity about what real a given integer is
mapped to, then a description like "the real number formed by changing the
nth digit of the binary expansion of the real number that maps to integer n
under function F" should itself be a well-defined finite description which
picks out a real number that differs from every number on the list by at
least one digit.
So, although the set of all well-defined finite descriptions must clearly be
"countable" in the traditional sense where arbitrary mappings are allowed,
it is not countable if only finite-describable mappings are allowed,
although it can easily be shown to be smaller than another countable set,
namely the set of all finite descriptions without regard for whether they
are "well-defined" or not (just list all strings of english symbols in
lexical order, although most of these will be completely meaningless like
"xhh{we2lkjk4j5j", and will thus not pick out any particular real number).
The basic problem here is that the notion of what it means for a description
to be "well-defined" can never be stated in terms of a formal rule--this is
reminiscent of the fact that no formal rules can encompass our understanding
of arithmetic, that no matter what axioms we use there will always be
theorems that we can prove to be true in ordinary-language proofs which
cannot be proved true or false by the axiomatic system. These
ordinary-language proofs will nevertheless be seen as quite rigorous by any
mathematicians who are Platonists about arithmetic--I believe Godel used
such a proof to show that the Godel statement G for the axioms of Peano
arithmetic (or any other axiomatization of arithmetic) must in fact be a
true theorem even though it's not provable using the axioms, as long as we
interpret the symbols to refer to our mental model of arithmetic (see the
discussion about Platonic 'models' in math at
http://www.mail-archive.com/everything-list.domain.name.hidden/thrd2.html#04463 for
more on this).
Jesse Mazer
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Received on Sat Nov 29 2003 - 16:39:26 PST