Re: Orlow cardinality question

Randy Poe said:
>
>
> Tony Orlow (aeo6) wrote:
> > Randy Poe said:
> > >
> > >
> > > Tony Orlow (aeo6) wrote:
> > > > Martin Shobe said:
> > > > > I don't simply declare omega to be the smallest infinity. In ZFC, it
> > > > > is a theorem that a smallest infinite ordinal exists. Once we have
> > > > > proven this, we name that smallest infinite ordinal, omega.
> > > > A theorem proven by what axioms? The smallest infinity is as nonsensical as the
> > > > largest finite.
> > >
> > > Here's where your intuition is correct:
> > > If we were to declare that there were a smallest infinite
> > > SET, then removing one element from that set would make
> > > a finite set. Removing an element creates a smaller subset.
> > Doesn't that translate into, "subtracting 1 from the smallest possible infinite
> > size of a set yields a finite size for the set"?
>
> No.
>
> Again, you are insisting on an axiom that doesn't exist,
> in this case that if A < B, then |A| < |B|. That is, if
> A is a proper subset of B, then the cardinality of A is
> smaller than the cardinality of B. As you have been told
> many times, there is no such requirement on infinite sets,
> and indeed a property of infinite sets is that there
> exists A < B with |A| = |B|.
Cardinality purports to describe the sizes of infinite sets. It certainly
generalizes from one approach to set size for finite sets. The distinction
between set size and cardinality is used when convenient. There isn't
originally supposed to be a difference, I mean, what is it that cardinality is
supposedly measuring if not set size? If you are not talking about the size of
the set when you say "infinite" or "finite" set, what ARE you talking about,
and if you say removing an element from a smallest infinite set yields a finite
set, then how is that different from saying that subtracting 1 from the
smallest infinite number yields a finite number? This distinction makes no
sense, and is worse than useless.
>
> > > Hence we must conclude that there is no such thing as
> > > a smallest infinite set.
> > Agreed.
> > >
> > > Where you are wrong is assuming that the cardinality of
> > > an infinite set is changed by removing one element. It
> > > is not. Your bigulosity has that property, so therefore
> > > we must conclude that there is no such thing as a
> > > smallest infinite bigulosity.
> > >
> > > But that does not impose a similar requirement on
> > > cardinality.
> > Well, this is one of the problems with cardinality,
>
> But the only "problem" is that it has a property you
> don't like. That's a very solipsistic view of "problem".
No, it's not a matter of fancy, but of consistency. If a system pretends to
measure set sizes, and we agree colloquially that removing an element from a
set results in a "smaller" set, but cardinality doesn't reflect this change,
then it is not doing the job it claims to be doing. It is not fully
distincuishing between sets of different sizes. This is fact, not a matter of
opinion.
>
> > and in my estimation, it
> > rests on an artificial distinction between numbers in general, and sizes of
> > sets.
>
> It is required by consistency.
No, it is required to maintain inconsistency, as I have been saying all along.
This distinction is a loophole that allows mathematicians to talk about set
sizes as if they weren't numbers, when in fact, that's exactly what they are,
and set and subset sizes are exactly what numbers represent.
>
> > Really, what does a number represent? It represents some number of
> > things, the size of the set of those things. We have 7 gallons of milk, or a
> > set of gallons, with a size of 7. We have 0.234 kilograms of gold. We have 234
> > elements in our set of grams of gold, which can apparently be broken into such
> > parts as reflected by the number that represents the set size. Whenever we are
> > talking about quantities using the abstraction of numbers, we are talking about
> > some measure of a set of units, or elements.
>
> This is all true for finite numbers. When we want to start
> axiomizing infinite things, we find that consistency requires
> that not all the intuitions about finite numbers can be
> imposed on infinite "numbers".
Each of those intuitions needs to be addressed, rather than simply dismissing
all intuitions wholesale. Ultimately, the best system is the one with the most
widely applicable rules, the rules that generalize most consistently from the
finite to the infinite. If you want to declare an intuition or general approach
to be untenable in the infinite realm, then a precise reason is in order.
Frankly, I am sick to death of hearing, "...because not every fact that holds
true for finite numbers or sets MUST hold true for infinite numbers or sets."
That is a non-answer to specific questions. I have not heard anyone explain
exactly which properties are maintained and which are violated, and how one
distinguishes clearly between those two sets of properties.
>
> The system you don't like is a very useful and completely
> self-consistent system. That there are different properties
> of infinite and finite numbers is a requirement of making
> the properties of infinite "numbers" be SELF-consistent.
> We have no known way to achieve both, and despite your
> insistences, neither do you. You have imposed a host of
> new inconsistencies.
Name one. I am resolving inconsistencies that exist despite all the denials in
behalf of Cantor. I offer you a system that maintains consistency and
distinguishes infinite sets in accordance with the intuitions that cardinality
violates, and it is rejected all around. I don't think the mathematical
community WANTS to have a system that deals consistently with the finite and
infinite. I think that upsets everyone's sense of awe. Oh well. It is easily
renewed by close examination of a snowflake or a spider web, or pondering the
very nature of matter. We don't need to surround the very idea of infinity with
artificial hocus pocus.
>
> - Randy
>
>

--
Smiles,

Tony
.

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