Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

From: Mike Oliver (mike_lists_at_verizon.net)
Date: 02/13/05 

Date: Sun, 13 Feb 2005 01:31:45 -0600

Keith Ramsay wrote:

> Let me confirm that I'm understanding you correctly.
> Is the idea that L(R) is relatively small among models
> of ZF containing R, and since it's thought to satisfy
> AD, it fails to satisfy CH? The rest was a stab at
> getting a model of ZFC (and hence not AD) that still
> doesn't satisfy CH, wasn't it?

Yes, that's right. The usual approach for
making L(R) into a model of ZFC is to force
over it to add a generic bijection between
R and aleph_1 (using countably closed forcing
so you don't add reals). But (i) that makes
CH true and (ii) I need a forcing that has
a generic object in V.

So my thought was to try forcing with partial
bijections between R and aleph_2 having cardinality
at most aleph_1. When forcing over a model
of choice, the last stipulation would make sure
cardinals were preserved up to aleph_2, and I
thought I might be able to argue from the existence
of R# or something that there was a generic in V.
But I don't think it's going to work, because there
just aren't any nontrivial conditions for the forcing
in L(R)--there are no subsets of R having cardinality
aleph_1. Well, it's too hard for this time of night.
But let me know if you think of anything.

Relevant Pages

  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... it fails to satisfy CH? ... over it to add a generic bijection between ... R and aleph_1 (using countably closed forcing ... in L--there are no subsets of R having cardinality ...
    (sci.math)
  • Re: Galileos Paradox
    ... this natural meaning should satisfy that A c B -> A is smaller than B ... Counting a finite set does ... Cardinality satisfies but not. ... It's a brutal reality. ...
    (sci.math)
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