Re: Orlow cardinality question

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"?
>
> 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, and in my estimation, it
rests on an artificial distinction between numbers in general, and sizes of
sets. 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. The distinction between the two,
and the different treatement each receives, leads to much of the inconsistency
in how set theory deals with questions of infinity.

When you say "Removing an element creates a smaller subset", why is that fact
NOT reflected in cardinality measures? When you say "Hence we must conclude
that there is no such thing as a smallest infinite set", doesn't that mean
there is no smallest infinite cardinality? If we want to bring these statements
into concurrence, don't we need to distinguish between oo and oo-1?
>
> - Randy
>
>

--
Smiles,

Tony
.

Relevant Pages

  • 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)
  • 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.physics)
Quantcast