Re: Cantor's Diagonal Argument
- From: "george" <greeneg@xxxxxxxxxx>
- Date: 3 Sep 2005 08:23:39 -0700
> I'm not sure what you mean here.
I mean that if you start with a countable set of numbers,
and use anti-diagonalization to generate more numbers to
add to the list, over and over again, ad infinitum,
"As long as you can", you will STILL never put an uncountable
number of numbers on the list. No matter how long you keep
expanding the list -- EVEN if you add-1 as transfinitely
many times as you like -- it will STILL be countable.
At least until you exhaust the countable ordinals.
THe possibility of which is not clear.
> Remember that Wittgenstein is not
> just denying the existence of "all real numbers".
Well, that would be bad enough. There is no point in
even going further if he is going to begin by denying
that much.
> He is denying the existence of "all cardinal numbers".
BU that is simply IRRELEVANT! One does NOT need to invoke
or reify "cardinal numbers" (or even numbers!) AT ALL, to
do this, unless one first knows how to defeat 1-1-correspondence
as relevant to size in general.
> "The symbol for a class is a list
In modern locution, that is just idiotic.
Most models have lots of elements that have
no symbols at all -- all non-standard models
of PA, for example.
> ... A cardinal number is an internal property of a list."
No, it isn't. It's just another list in its own right.
If we want to use cardinal numbers to talk about the sizes
of classes then the cardinal number is an EXternal property
of the CLASS, NOT any property at all (except by association)
of the list.
> "There's no such thing as "all numbers" simply because there are
> infinitely many."
And that is just ridiculous.
This is a denial of the existence of even ONE real number.
IT makes about as much sense as saying that there's no such
thing as pi or e simply because their decimal expansions are
are equally long. This is about denying the existence of infinite
sets period. If that's really what you want to do, then just
delete ZF's axiom of infinity and operate with your favorite finite
set theory. That is NOT problematic, logically OR philosophically!
What IS problematic philosophically is alleging that "the real world"
obeys some sort of set theory that just happens to be finitary
(which is basically what is going on here). I can't imagine that
anyone cares, frankly.
> W would probably question even the distinction between
> "set" and "class". However, he was writing in the late 1920's when
> this distinction was (I think) not available.
In the context of the classical first-order paradigm,
the distinction is unavoidable. Your model is going to
be whatever size it is. Its every subcollection of that same maximal
size
MUST be a proper class, in the opinion of that model. You DON'T
get to have an OPINION about that, withIN that family of set theories.
VonNeumann proved this around 1927 so, yes, W might not have seen
it yet. But I doubt he would've cared; he was denying much more
basic things already.
>
> > and to the idea we can pretend to compare the "set"
> > of real numbers in magnitude with that of cardinal numbers.
But why do we even NEED "magnitudes" at all?? Isn't that
JUST an alias, a euphemism, FOR "cardinal number of"?
THere is simply no difference -- beyond a redundant layer
of indirection between "the magnitude" of a set and "the
cardinal number of" the same set. Measuring "magnitudes"
was ALWAYS, DEFINITIONALLY, what cardinal numbers WERE FOR!
Nobody was ever trying to talk about the "magnitude" of the
class of "all" cardinal numbers! Everybody had known SINCE
THE BURALI-FORTI PARADOX, 30 years prior, that you just plain
DON'T GET TO HAVE a class of all cardinal numbers (what would
ITS cardinality be??). If the class cannot exist at all
then comparing its "magnitude" to that of any other class
is precluded.
>
> If reals and
> cardinals both exist at all then there are certainly more
> cardinals than reals.
> ---
>
> No, if you are using "more" in the sense of standard English.
No, YES, if you are using "more" in the sense of standard
English. ANd since W is not conceding that either of these
classes exists at all, I don't see why it matters.
>
> "What we call the "correlation of all the members of a class with
> others" in the case of a finite class is something quite different from
> what we, e.g., call a correlation of all cardinal numbers with all
> rational numbers. The two correlations, or what one means by these
> words in the two cases, belong to different logical types. An infinite
> class is not a class which contains more members than a finite one, in
> the ordinary sense of the word "more".
He really is just plain lying here.
If BOTH sets are infinite then we can allege that "the logical more"
is different from the natural one. But if one set is finite and
the other is infinite, then we cannot. The same old usual
natural-language definition works properly in the infinite>finite
case BECAUSE IT WORKS EXACTLY THE SAME way as it works in the
finite>finite case. W is under obligation to expand HIS definition
of the finite>finite case. Once he does, the fact that it applies
identically/analogously in the infinite>finite case will be immediately
demonstrable.
> If we say that an infinite
> number is greater than a finite one, that doesn't make the two
> comparable,
Of course it does; indeed, we would never have KNOWN THAT
the infinite set WAS infinite IF they weren't comparable!
> because in that statement the word "greater" hasn't the
> same meaning as it has say in the proposition 5 > 4
I repeat, that is simply a lie; it DOES SO TOO have the
EXACT same sense. The DIFFERING sense comes in AND ONLY
in the infinite>INfinite case.
>
> "The form of expression "m=2n correlates a class with one of its proper
> subclasses" uses a misleading analogy to clothe a trivial sense in a
> paradoxical form.
He can't support this. Yes, we do this.
But note that this is the infinite>INfinite case AND NOT
the infinite>finite case. He still has FAILED his burden of
impeachment of THAT case, totally.
More to the point, he acts as though our natural language
here could possibly be misleading by analogy. That is NOT possible.
The arena where this correlation is fundamentally transpiring
IS NOT in natural language!
The existence of this correlation is a syntactic artifact of a
syntactically defined first-order theory! It is NOT legitimately
disputable! LET ALONE paradoxical! LET ALONE capable of "misleading"
anyone! Or even of BEING "analogous"! It is simply a theorem!
> (And instead of being ashamed of this paradoxical
> form as something ridiculous, people plume themselves on a victory over
> all prejudices of the understanding).
The burden of proof of existence of paradox is on HIM, and
we are NOT holding our breath. Natural-language attacks against
proofs are quixotic.
> It is exactly as if one changed
> the rules of chess and said it had
> been shown that chess could also be
> played quite differently."
That flies only if he can articulate the original rules.
In the finite>finite case, proper subset/superset sufficed to
force difference-of-size. In the infinite>finite case, that is
STILL the case; that case is NOT different. In the infinite>infinite
case, this criterion starts to fail. If the other side thinks
it can articulate a coherent theory of size while preserving
this measurement, it is welcome to try. BUt it's hopeless.
Precisely as they (self-REFUTINGly, though they didn't see it)
argued, the criteria (proper part vs. 1-1-correlation) BIfurcate,
for infinite sets. They DON'T bifurcate for finite sets.
This proves by THEIR lights that the finite and the infinite
are different in some deep important way. To say that this way
is "well, the infinite sets don't really have sizes, and can't
be compared in size, at all" is, as I said, SELF-refuting, because
THEY were the ones who insisted that proper-part-hood DID support
comparsion in size. Finite sets, though, can be compared in
size both when they are disjoint and when their intersection is
a proper subset of both sets. Infinite sets, again by THEIR
lights, cannot. This AGAIN implies that THEY think finite is
different from infinite on some deep level. Their claim that
this deep difference IS NOT one of "size" is, I again repeat,
belied by the fact that THEIR definition WORKS NORMALLY
in the infinite>finite case.
> He is not disputing Cantor's theorem. He is denying that the
> expression "the reals" have the meaning you want it to mean.
That is completely ridiculous. We have an axiomatic
framework. We DEFINE the reals. In the context of our
axioms, EVERYthing means EXACTLY What WE say it means.
If he wants different meanings then he has to come up
with different axioms OF HIS OWN.
> ---
> Why would you want to be airing the man's dirty
> laundry? You don't really expect this to have a
> larger effect on people's opinions about set theory
> than it has on their opinions of Wittgenstein,
> do you?
> ---
>
> Perfectly good laundry, in my view. As Ray Monk has pointed out, W's
> philosophy of mathematics is central to his whole philosophy.
Hardly. W did NOT HAVE a philosophy of math that would be
recognized as such in modern terms. The classical results
MUST be accomodated.
.
- References:
- Cantor's Diagonal Argument
- From: William of Ockham
- Re: Cantor's Diagonal Argument
- From: george
- Re: Cantor's Diagonal Argument
- From: William of Ockham
- Cantor's Diagonal Argument
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