Re: Orlow cardinality question



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|.

> > 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".

> and in my estimation, it
> rests on an artificial distinction between numbers in general, and sizes of
> sets.

It is required by consistency.

> 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".

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.

- Randy

.

Relevant Pages

  • Re: Dial 999 for the real number line
    ... Alternatively, the meaning of an infinite decimal expansion, ... The whole idea of cardinality, for example, is rubbish. ... maths is not physics. ... it's necessary to start with a set of axioms. ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... >>And then at some point this led us to deciding that cardinality and ... >>meant for an infinite set to have a certain number of elements. ... always goes hand in hand with subtraction. ... Does the interval on the reals have to include 1? ...
    (sci.math)
  • Re: Cantors diagonal proof wrong?
    ... >> So you claim that all infinite sets have the same cardinality? ... >What I don't understand is what the notion of cardinality really is. ... on about, it's not mathematics, it's a confession of faith (and ... or on whether "axioms" in one system or another ...
    (sci.math)
  • Re: An uncountable countable set
    ... You agreed that it is not a cardinality ordering. ... 'infinite') in one ordering, which is not a cardinality ordering, is ... problem is that you spout about non-standard analysis while you haven't ... and mathematics that that book uses as that book is written ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... all Cantorian approaches like this, that they are in the same general ... >>Not contradiction according to Cantorian cardinality. ... Why should it not be true of infinite ... And what finer distinction does it make than between sets that are ...
    (sci.cognitive)