From: Brandsma
Subject: Re: well-orderings in R( was Re: Continuum Hypothesis Question)
Date: Thu, 14 Jan 1999 17:07:58 +0100
Newsgroups: sci.math
Keywords: Countable ordinals embed in R
feldmann4350@my-dejanews.com wrote:
> I have now a slightly different question: is it true that any countable
> ordinal can be immersed in R ,ie is there an increasing fonction from alpha
> (the ordinal) to R (or Q)with usual ordering?
Yes, this is true. It can be proved by induction e.g., or by using the fact that
they are 0-dimensional separable metric spaces in the order topology, and the
Cantor set (which is also an ordered space) is universal for those. So even in
the (standard) Cantor set they can be embedded.
> This is (almost) obviously
> equivalent to "any countable limit ordinal alpha is limit of an increasing
> sequence alpha_1, alpha_2,...,alpha_n,..." (is there a notion of weak
> cofinality covering this?), but this last result (which i thought obvious at
> first)seems hard to prove (or am i missing something?)
This is indeed also true: let f be a bijection from omega onto alpha. Let alpha_1
= f(1), and pick alpha_(n+1) > max( alpha_n, f(n)), which can always be done,
as alpha is a limit. Because f maps onto alpha, the alpha_n 's must be cofinal.
Henno Brandsma
==============================================================================
From: ikastan@sol.uucp (ilias kastanas 08-14-90)
Subject: Re: well-orderings in R( was Re: Continuum Hypothesis Question)
Date: 14 Jan 1999 21:44:40 GMT
Newsgroups: sci.math
In article <77ku09$vv0$1@nnrp1.dejanews.com>,
wrote:
@In article <77jbju$7fa$1@hades.csu.net>,
@ ikastan@sol.uucp (ilias kastanas 08-14-90) wrote:
...
@> We cannot... not even in ZF + AC + GCH. Such an f would be
@> a definable wellordering of an uncountable set of reals; there is a model
@> where any definable wellordering of a set of reals is countable...
@>
@I have now a slightly different question: is it true that any countable
@ordinal can be immersed in R ,ie is there an increasing fonction from alpha
@ (the ordinal) to R (or Q)with usual ordering? This is (almost) obviously
Yes; any countable linear ordering can be realized as a subset of
Q (in fact, any countable topological space is homeomorphic to some subset
of Q). Anyway... a bijection w <-> alpha exists by definition, i.e.
an enumeration b_0, b_1, ... of the ordinals < alpha. Assign to b_0
your favorite rational, q_0. Pick a q_1 > [or < ] q_0 if b_1 > [<] b_0.
Pick q_n+1 to reflect the position of b_n+1 relative to b_0, ..., b_n.
That's all.
@equivalent to "any countable limit ordinal alpha is limit of an increasing
@sequence alpha_1, alpha_2,...,alpha_n,..." (is there a notion of weak
@cofinality covering this?), but this last result (which i thought obvious at
@first)seems hard to prove (or am i missing something?)
Let b_0 be alpha_0; alpha_1 = the first b_j > alpha_0... then
the first b_k > alpha_1 will be alpha_2, and so on.
Ilias