Re: Orlow cardinality question

Randy Poe said:
>
>
> Tony Orlow (aeo6) wrote:
> > Cardinality purports to describe the sizes of infinite sets.
>
> No it doesn't. Tony purports to ascribe this meaning to
> cardinality.
Then why am I told I am wrong when I say there are half as many even integers
as all integers, based on cardinality? You are being disingenuous here.
Cantorians consistently claim that they are measuring the sizes of infinite
sets with absolute surety, and then try to back out of that claim to save their
system, at which point they can't justify the system, because it has no purpose
if measuring sizes of sets isn't it.
>
> > It certainly
> > generalizes from one approach to set size for finite sets.
>
> Yes, it does. And when you generalize, you may change some
> properties.

Yes, but they should be generalized to they apply to all cases, not replaced
with exceptions and dismissals. That's not generalization.
>
> > The distinction
> > between set size and cardinality is used when convenient.
>
> No, it's used always.
And what IS the distinction, when it comes to infinite sets? That there is no
such thing as set size, even though the set contains elements, so one exists
and one doesn't? If that's your distinction, it's bogus.
>
> > There isn't
> > originally supposed to be a difference,
>
> This is a Tony Declaration, an additional imposition made
> on the system. You do a lot of those.
>
> There isn't any "originally" because we start with no
> well-defined notion of "size" for infinite sets. Given that,
> there's nothing for us to match our cardinality up with
> to see if it's the same.

Cardinality IS set size for finite sets, and is generalized to infinite sets to
extend this notion of size measure. If you disagree with this statement, then
tell me what the impetus for developing cardinality measures for infinite sets
was. Be honest. I am not making this up. That was the intent, and is still the
claim.

>
> > I mean, what is it that cardinality is
> > supposedly measuring if not set size?
>
> It is what it is: a complete ordering based on ability
> to biject. It is a natural generalization based on how
> we compare finite sets. The fact that it is a COMPLETE
> ordering is very useful, and very important. It means
> that there is a way to compare any two sets regardless
> of the type of contents. Subset-based attempts are at
> best partial orders.

A complete ordering? It does not even differentiate between the sizes of a set
and its proper subset! Sure you can compare them. They're the same! How useful!
It doesn't matter that one is exactly half the other. They're the same size!
That doesn't sound very complete in my book. Of course, your book has a
different definition of "complete" when it comes to orderings, but maybe it's
not a very complete definition.

>
> > If you are not talking about the size of
> > the set when you say "infinite" or "finite" set, what ARE you talking about,
>
> Bijection.
>
> Specifically, ability to biject with subsets of the naturals.

Okay, fine, if you want to back into that corner, and stop claiming that
cardinality IS the set size for infinite sets, then play with your bijections
all you want. Personally, I am a tad more interested in actually studying
infinities in as much detail as possible and developing better methods for
dealing with zeroes and infinities than what is currently available.

>
> > and if you say removing an element from a smallest infinite set
> > yields a finite set,
>
> Of course I never said that. I believe I said the
> opposite.

I believe you said that you can never have a smallest infinite set, because
that would mean that removing an element would produce a finite set, since
removing an element always makes a somehow smaller set. Of course, it would be
easier to tell if you hadn't snipped the context. It's really hard to pin down
what the Cantorians really think. It's a moving target. Did you say you cannot
have a smallest infinite set, but you can have a smallest infinite set size?

>
> > then how is that different from saying that subtracting 1 from
> > the smallest infinite number yields a finite number?
>
> The subtraction processes are different.

Subtraction is different from what happens to set size when elements are
removed? Pray tell, what is the difference?

>
> One way to think of it:
> When I subtract an element from an infinite set, I don't
> subtract 1 from the cardinality.

Interesting. It seems like making the set one element smaller would reduce the
set size by 1. But, I guess cardinality of infinite sets is unrelated to set
size.

>
> Another way to think of it:
> Subtracting 1 from an infinite cardinality doesn't change
> the cardinality.
Same thing. This is just a convenient rule to allow a smallest infinity, and
really a cheat.
>
> At any rate, I don't claim in either case that you can
> subtract 1 from something infinite and get something
> finite. Both claims are false and opposite to what I
> did say.
Then there is no smallest infinity, such as omega, any more than there is a
largest natural number, say, alpha. That was my point, among many.

>
> What I *did* say is that there's a smallest infinite
> number, but no smallest infinite set. Do try to argue with
> what I am saying rather than changing my words. The
> reason there's no smallest infinite set is that you can
> always remove an element from an infinite set and
> have another infinite set. The reason there can be a
> smallest infinite number is that a set whose cardinality
> is alpha, still has cardinality alpha even when you
> subtract an element.
Yes, like I said, the fact that adding or subtracting elements doesn't change
an infinite cardinality is a convenient method for allowing a smallest
infinity, while disallowing a largest finite natural, which is yet another in
this monkey's fist of inconsistencies and exceptions. Sorry, but I have already
shown how one can count backwards from infinity and perfom the subtraction,
while never reaching the finites, and how there is no smallest infinity. This
is a kludge.
>
> > This distinction makes no sense
>
> To you.
>
> > and is worse than useless.
>
> To you. But it is enormously useful in fact.

In fact it is alarmingly damaging, discombobulating people's logic and standing
in the way of productive development in this area.

>
> - Randy
>
>

--
Smiles,

Tony
.

Relevant Pages

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