Re: Aleph One Sets

From: Mike Oliver (mike_lists_at_verizon.net)
Date: 10/22/04 

Date: Fri, 22 Oct 2004 12:08:38 -0500

Keith Ramsay wrote:
> Daryl McCullough <daryl@atc-nycorp.com> wrote:
> |Define an equivalence class on sets of reals as follows: R1 ~~ R2 if
> |there exists an order-preserving bijection between R1 and R2. What is
> |the cardinality of the set of equivalence classes of ~~?
>
> In article <2tnjt2F225n3jU1@uni-berlin.de>, Mike Oliver
> <mike_lists@verizon.net> writes:
> |Here's what I'm wondering: Can you actually embed 2^{2^aleph_0}?
> |I don't immediately see why or why not.
>
> I have a simpler argument that the answer is yes.
>
> There are 2^c sets of reals that are dense in the reals. The
> order-preserving bijections between dense sets of reals are
> all restrictions of continuous functions from the reals to
> the reals. Since continuous functions can be determined by their
> values on the rationals, there are c^{aleph-0} = c continuous
> functions. If x<=2^c and x*c>=2^c, it must be that x=2^c. I
> don't know whether I needed the axiom of choice here.

For this argument you do, yes. Essentially you're arguing
that each Wadge class has cardinality at most c and there
are 2^c representatives, so there must be at least 2^c
Wadge classes.

But in a model of AD, the Wadge classes can be wellordered
whereas P(R) certainly cannot, so there can't possibly
be an injection from P(R) into the collection of Wadge classes.

Relevant Pages

  • Re: Aleph One Sets
    ... Mike Oliver wrote: ... >> all restrictions of continuous functions from the reals to ... Since continuous functions can be determined by their ... > But in a model of AD, the Wadge classes can be wellordered ...
    (sci.math)
  • Re: Aleph One Sets
    ... Mike Oliver wrote: ... >> all restrictions of continuous functions from the reals to ... Since continuous functions can be determined by their ... > But in a model of AD, the Wadge classes can be wellordered ...
    (sci.logic)
  • Re: Aleph One Sets
    ... |there exists an order-preserving bijection between R1 and R2. ... There are 2^c sets of reals that are dense in the reals. ... order-preserving bijections between dense sets of reals are ... Since continuous functions can be determined by their ...
    (sci.math)
  • Re: Aleph One Sets
    ... |there exists an order-preserving bijection between R1 and R2. ... There are 2^c sets of reals that are dense in the reals. ... order-preserving bijections between dense sets of reals are ... Since continuous functions can be determined by their ...
    (sci.logic)
  • Re: Aleph One Sets
    ... > There are 2^c sets of reals that are dense in the reals. ... > order-preserving bijections between dense sets of reals are ... Since continuous functions can be determined by their ... But in a model of AD, the Wadge classes can be wellordered ...
    (sci.math)