Re: Proposed definition for comparing the sizes of two sets

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 11/29/04 

Date: Sun, 28 Nov 2004 23:42:12 -0800

Virgil wrote:

> In article <4b0c747b.0411282057.3a22c922@posting.google.com>,
> dchris@netcom.ca (Dan Christensen) wrote:
>
> > Proposed Definition: A set Y is said to be larger than a set X iff there
> > exists no function mapping X onto all of Y. (i.e. there is no surjection
> > from X to Y)
> >
> > Is this a workable definition that covers all cases? If so, is it widely
> > used?
> >
> > Dan
>
> It is essentially the definition Cantor used, and I believe it covers
> all cases provided one has the axiom of choice.

There are definitions of relative set sizing besides cardinality, for
example, there is a definition of set sizing that a proper superset of a set
is larger than the set, for finite and infinite sets, as verified by Katz.
As well, in the consideration of number theory, the set of even integers, as
an example, has half the asymptotic density of the integers, within the
integers. Another notion of larger is X > Y when X has infinitely many
proper subsets that are proper supersets of Y, essentially a stronger
condition than the proper superset size relation.

The asymptotic density is rather useful, for example, half of the integers
are even integers, and twice as many integers are not multiples of three as
are multiples of three.

For example, consider the power set of the naturals where half of the subsets
of the naturals have as an element a given element of the naturals, and a
quarter have both of two elements. The set of all subsets of the naturals
containing as an element a given element of the naturals is "half" the size
of the entire set.

One way to reinforce that intuitive concept is as so: compare 1/2 to 1/4.
Pick a subset of the set of all natural numbers at random, the chance that it
contains as an element zero is one half, and that it contains 18 zillion even
is one half, and that it contains both is one fourth.

For disjoint sets, the above methods can work by comparing each to their
union, or some other superset of their union, and then comparing those
relative sizes.

There are probably yet other ways to meaningfully compare two sets' sizes.
Using cardinality will probably get you a passing grade. There is
considerable argument about whether infinite sets automatically have
bijections.

Ross F.

Relevant Pages

  • Re: infinity
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  • Re: Galileos Paradox and the Project of the Reals
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  • Re: Galileos Paradox and the Project of the Reals
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  • Re: Galileos Paradox and the Project of the Reals
    ... call cardinality "size" becase it makes intuitive sense to most people ... It uses it as shorthand for bijectability. ... If I make statements about sizes of infinite sets and am told I am wrong because I am contradicting set theory, ... Each such function can be defined by the natural mapped to each natural, and so consists of a countably set of naturals. ...
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  • Re: Galileos Paradox and the Project of the Reals
    ... call cardinality "size" becase it makes intuitive sense to most people ... It uses it as shorthand for bijectability. ... If I make statements about sizes of infinite sets and am told I am wrong ... and so consists of a countably set of naturals. ...
    (sci.math)
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