Re: Tarski's weird definition of cardinal numbers
Re: Tarski's weird definition of cardinal numbers
Source: http://sci.tech−archive.net/Archive/sci.logic/2005−10/msg00146.html
• From: "Rupert"
• Date: 4 Oct 2005 23:08:15 −0700
andrewspencers@xxxxxxxxx wrote:
> The usual definition I've seen of a cardinal number x is the class of
> all cardinal numbers less than x, with the cardinal number 0 being the
> empty class.
This isn't really satisfactory as it stands, because no definition is
given of "less than". There's a version of this definition that can be
made precise, but it's a definition of ordinal rather than cardinal
numbers.
> Thus all cardinal numbers are finite classes,
Why? What about w={0,1,2,...}?
> and the
> ordering relation is the subset relation. The cardinal number of a
> finite class K is the cardinal number which is equinumerous with K, and
> the cardinal number of a cardinal number x is x itself.
>
> But Tarski says on p. 80 of "Introduction to Logic", "The cardinal
> number of a class K is the class of all classes equinumerous with K."
> Therefore all cardinal numbers are infinite classes, with the exception
> of the cardinal number 0, which is a class containing one element (that
> element being the empty class), i.e. the cardinal number 0 is the class
> {{}}, since the empty class is the only class equinumerous with the
> empty class. The cardinal number 1 is the infinite class containing all
> of the following classes: {{}}, {{{}}}, {{{{}}}}, etc, as well as
> {{{},{}}}, {{{},{},{}}}, {{{},{},{},{}}}, etc, since those classes each
> have exactly one element. The cardinal number of a cardinal number x is
> not x itself, but some other infinite class. Besides all that, the
> ordering relation is nowhere in sight.
>
You could say the cardinal number A is less than the cardinal number B
if there's an injection from a member of A into a member of B, but no
bijection from any member of A to a member of B.
> Who's responsible for perpetrating that latter definition (Tarski
Re: Tarski's weird definition of cardinal numbers 1
Re: Tarski's weird definition of cardinal numbers
> himself?), does anybody still use it today, and what advantage does it
> have over the former definition?
The definition originated with Frege. The main problem with it is that
in most modern theories of sets there's no such thing as the set of all
sets equinumerous with a given set. In some theories of sets there are
sets and classes, where every set is a class but not the converse, and
only sets can be members of a class, and in that case the definition
will work okay if we define a cardinal number to be the class of all
sets equinumerous with a given set. The cardinal number will then not
be a set, it will be a proper class (a class which is not a set). Then
the problem with that is that cardinal numbers can't be members of
sets, and we sometimes want them to be. The way to get around this is
Scott's trick. Ordinal numbers can be defined to be sets which are
transitive and well−ordered by the membership relation. (This is the
precise version of your definition I was alluding to earlier). Then you
can associate an ordinal number to each set called its rank. The empty
set has rank 0, and the rank of any set is the first ordinal greater
than the ranks of all its members. Then you can define a cardinal
number to be the set of all sets equinumerous with a given set that are
of least possible rank among the sets equinumerous with that set. This
will be a set rather than a class. This is the usual definition of
cardinal number when we don't assume the axiom of choice. If we do
assume the axiom of choice, then we can define ordinal numbers in the
way I have mentioned and define cardinal numbers to be ordinals not
equinumerous with any smaller ordinal. (We will then need the axiom of
choice to prove that every set is equinumerous with a unique cardinal).
The advantage that Tarski's definition has over your "definition" is
that it's a definition. You didn't really give a definition. You might
have been thinking of the definition of ordinals as transitive sets
well−ordered by the membership relation and cardinals as ordinals not
equinumerous with any smaller ordinal; that's the usual definition
today. Then Tarski's definition has the advantage over this definition
that you don't need to assume the axiom of choice to prove that any set
has a cardinal, but the disadvantage that you need to work with proper
classes and that your cardinals can't be members of sets. However, this
disadvantage can be remedied with Scott's trick.
.
• Follow−Ups:
♦ Re: Tarski's weird definition of cardinal numbers
◊ From: Ross A. Finlayson
♦ Re: Tarski's weird definition of cardinal numbers
◊ From: andrewspencers
• References:
Re: Tarski's weird definition of cardinal numbers 2
Re: Tarski's weird definition of cardinal numbers
♦ Tarski's weird definition of cardinal numbers
◊ From: andrewspencers
• Prev by Date: Lowenheim−Skolem−Ockham Theorem
• Next by Date: Re: Skolem Again
• Previous by thread: Tarski's weird definition of cardinal numbers
• Next by thread: Re: Tarski's weird definition of cardinal numbers
• Index(es):
♦ Date
♦ Thread
Re: Tarski's weird definition of cardinal numbers 3