Re: Anti-diagonalist page

On 5 Sep 2005 20:33:33 -0700, slaterbh@xxxxxxxxxxxxxxxxxx wrote:

>This seems to be last last post to date of Ullrich, who has also, by
>now, calmed down enough to stop using big capitals,

You should pay more attention to who says what - the only
capitals in my posts are at the start of sentences. And
perhaps in other traditionally capitalized places, but the
person who tends to write in CAPITALS because he thinks
we're deaf is not me.

>though not, I
>notice, terms of abuse. A little more time in a cooler atmosphere, and
>he might even begin to see I am right.

I doubt it. Let's ignore things like the idea that there's something
paradoxical about Skolem, there are much more basic problems.
You say

"Hence such 'sets' are not
numerable, and one-one correlation does not equate with equal
numerosity, as Hume's Principle supposes."

But in mathematics the _definition_ of "equal numerosity" is
the existence of such a correlation.

Then equiniumerosity is an equivalence relation. These
days people tend to define an infinite cardinal as one
particular member of an equivalence class wrt this reliation.
That requires some set theory - if instead we simply say that
a cardinal _is_ one of those equivalence classes things are
simpler (in some ways) - now the fact that there exist
non-equinumerous infinite sets _does_ show that there exist
unequal infinite cardinals, by definition.

There exist plenty of people who insist that the proof
that there is no bijection between the reals and the
natural numbers is wrong. I gather you're not one of
them, great. But the only hypothsis I'm left with is
that your point is that the standard definitions in
mathematics are somehow _wrong_, and this is simply
incoherent, a definition cannot be wrong, a definition
simply says what a certain term shall mean.

Someone else has pointed out that we cannot conclude
that we've proved that there exist infinite numbers of
various sizes, if "number" means what it does in
ordinary speech. The reply to that is well duh, of
course "infinite cardinal" does not mean the same as
what "number" means in ordinary speech, nobody ever
claimed it does.

>The original post from 'William
>of Ockham' gave the web address of an article of mine, also a short
>excerpt. Disucussion has taken off as though the excerpt was all I
>said. Not so. And subsequently I have given another details of a
>paper by writer who mounted the same defense of Finitism (yes:
>Finitism!), Crispin Wright. It's time for some quiet reading and
>reflection, I think.

Ignore the following paragraph - it's motivated largely
by your complaint about "insults". The math resumes in
the paragraph after the following:

You think so? Nobody could possibly think that you're
simply wrong if he'd reflected on these things?
The reason that essentially every mathematician on
the planet disagrees is solely due to a lack of
reflection? Right. Got it.

But seriously: The idea that you're defending finitism
was not at all clear to me from that excerpt. If you
read the thread instead of focussing on the fact that
people are daring to say you're wrong you'd have noticed
me saying that if one insists that there is no such
thing as infinite sets in the first place that's not
something I'd regard as simply ludicrous (whether
that's exacyly what you mean by "finitism" is not
clear to me...)

But you still _seem_ to be unaware of actual
mathematical practice. You say "only the presumption
that there are infinite numbers can turn whatever
impossibility there is here into a seeming demonstration
that the number of the real numbers is greater than
the number of natural numbers". I'm not sure whether
what we're objecting to here is the presumption that
there exist infinite _sets_, or the idea that some
infinite things deserve to be called infinite
_numbers_. If the first: Mathematicians do not
"presume" that infinite sets exist - the existence
of an infinite set is an _explicit_ axiom of set
theory. The meaning of the word "axiom" has changed
since Euclid: an axiom is no longer a self-evident
truth, it's something that we _assume_ in a given
context, for the sake of working out what follows.
If we were simple "presuming" that infinite sets
exist then we wouldn't feel the need to insert
an explicit axiom of infinity into set theory.

If otoh, as I thought was clear until you said
you were defending finitism, you have no problem
with infinite sets but your point is that we must
not refer to infinite numbers, well again that's
just silly, certain things are called infinite
cardinals just by definition.

************************

David C. Ullrich
.

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