Re: Set Theory
From: KRamsay (kramsay_at_aol.com)
Date: 11/21/04
- Next message: C. Bond: "roots vs zeros"
- Previous message: KRamsay: "Re: Is this math test too easy?"
- Next in thread: Stephen J. Herschkorn: "Re: Set Theory"
- Reply: Stephen J. Herschkorn: "Re: Set Theory"
- Reply: Stephen J. Herschkorn: "Re: Set Theory"
- Reply: J.E.: "Re: Set Theory"
- Messages sorted by: [ date ] [ thread ]
Date: 21 Nov 2004 20:16:47 GMT
In article <W2Pnd.4251$c21.1773@news02.roc.ny>, "Steven"
<sgottlieb60@hotmail.com> writes:
|I just learned that aleph null < 2^ aleph null = c < 2^ c< (2^c)^c ... Has
|anyone found any cardinalities in between yet??
The infinite cardinalities in order of size are denotes aleph_0,
aleph_1, aleph_2, and so on. They correspond to the ordinals,
including infinite ordinals, so there is a smallest cardinality
greater than aleph_n for natural numbers n, called aleph_omega.
The cardinality aleph_1 is the cardinality of the set of countable
ordinals. These are order types of well-ordered countable sets.
A total ordering < on a set X is called a well-ordering if each
nonempty subset of X has a least element in the ordering <.
Two well-ordered sets are considered to have the same order
type if there is a one-to-one order-preserving correspondence
between them.
Since we only care about the order type, we can represent
each countable ordinal as an ordering on the natural numbers
(not necessarily the same as the usual ordering, of course).
This can be encoded as a subset of NxN, i.e. the pairs (m,n)
where m is before n in the well-ordering. Pairs of natural numbers
can be encoded as natural numbers. So each well-ordering
can be represented as a subset of the natural numbers. There
will be many different encodings that have the same order type,
but since we're assuming the axiom of choice, we conclude that
there exists a set of real numbers that is in a one-to-one
correspondence with the countable ordinals, which have
cardinality aleph_1.
If there is any cardinality between aleph-null and c, then aleph_1
is a cardinality between aleph-null and c. So when people ask
whether there are cardinalities between aleph_0 and c, the problem
is not with "finding" the cardinality; the problem is with whether a
certain cardinality we know about already is less than c.
The continuum hypothesis says that c and aleph_1 are the same,
which would mean that there is no cardinality intermediate between
aleph_0 and c. The continuum hypothesis is equivalent to saying
that there is a well-ordering of the real numbers in which the set of
reals less than a given real in the well-ordering is countable. That's
equivalent to various other odd things, like the existence of a set
in the xy plane whose intersection with each line parallel to the x
axis is countable, and with a completement whose intersection with
each line parallel to the y axis is countable too. If we have a well-
ordering <' on the reals for which {s: s<'r} is countable for each real r,
the set {(r,s) : r <' s} is such a set.
Goedel suspected at one point that the continuum is aleph_2, and
I believe there are others who think there may be reasons to suspect
that.
The problem of course is as other people have pointed out that
the axioms we currently use don't decide the question. There is a
proof that c is not the sum of countably many smaller ordinals, so
c can't be aleph_omega (which is the sum of aleph_n for natural
numbers n). We can also say that c is not aleph_{2^c} and some
other things defined in terms of c, that by the way they are defined
have to be larger than c. There is a sense in which one can assume
it is anything else, however, and still remain consistent with ZFC,
a commonly used set of axioms for set theory (assuming ZFC is
itself consistent).
People have searched for new axioms that would both make
intuitive sense, and answer the question. They've succeeded to some
degree with finding plausible new axioms. (The extent to which they
have succeeded is a matter of judgement; some people are quite
happy with a lot of the new axioms, though.) There are so-called
"large cardinal" axioms. But they don't decide the continuum
hypothesis.
As a result, people usually think one of two things, either that
the continuum problem is hard, or that it's not a problem
to solve. Platonists think of the problem as hard. It's a theorem
of ZFC that c=aleph_x for some ordinal x. A Platonist would
(ordinarily at least) believe that, and think that we just don't know
yet which ordinal x is, whether it's 1, 2, or something else. On
the other hand, there are people who think that the question is
somehow undefined, and that it's meaningless to ask which
ordinal x "really" is.
Keith Ramsay
- Next message: C. Bond: "roots vs zeros"
- Previous message: KRamsay: "Re: Is this math test too easy?"
- Next in thread: Stephen J. Herschkorn: "Re: Set Theory"
- Reply: Stephen J. Herschkorn: "Re: Set Theory"
- Reply: Stephen J. Herschkorn: "Re: Set Theory"
- Reply: J.E.: "Re: Set Theory"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
