Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/21/05 

Date: 21 Jan 2005 11:51:10 -0800

H. Enderton wrote:
> Ross A. Finlayson <raf@tiki-lounge.com> wrote:
> >In ZFC, there is at least one well-ordering of the reals.
>
> Yes. But it is a theorem that, working in ZFC, you cannot
> give a specific *definable* example of one. That is, for any
> formula that you think might define a well-ordering of the reals,
> there is a model of ZFC in which it doesn't. (A result due to
> Sol Feferman, back in the early days of forcing.)
>
> To digress, there is a definable well-ordering of the *constructible*
> reals (i.e., the reals in Goedel's class L). But that's only a drop
> in the bucket.
>
> --Herb Enderton

Hi,

Thanks, Herb. I think you can provide a lot of insight into this.

Forcing, or coconsistency, dependent consistency, is the notion that
some things are proved relative to the consistency of their theory.
That is, both are consistent, or neither, for example Con(ZFC) <=>
Con(ZFC + IST), where IST is Nelson's Internal Set Theory. A large
collection of those results would be decided were ZFC shown
inconsistent. ZFC is generally assumed to be consistent.

The "reverse mathematics" idea is to use the _least_ set of axioms to
prove a result. For example, instead of using all of the axioms ZFC,
only use the axioms you need. That kind of axiomatization basically
would help limit the repercussions of Not Con(ZF), where some subset of
the axioms of ZF would probably remain consistent, _were that to be the
case_.

Dr. Enderton, I hope you would explain some more of the meaning of
Feferman's result about well-ordering the reals. We could ask him.

http://math.Stanford.EDU/~feferman/

It would probably be a good idea if I wanted to know to research him
before contacting him. Then, I'd invite him to come inveigh on
sci.logic.

Herb, I think V = L. So do some others, so that is not quite so awash
as claiming infinite sets are equivalent in terms of consensus, a
plurality of people agree that V = L, that the universe is the
constructible universe. Does that apply to the Goedelian-constructible
reals? Does Feferman's result only thus apply if V =/= L?

For the constructible case, you relate that there is a "definable"
well-ordering: is there an example?

I was interested to read some of the descriptions on the web page for
the UCLA Logic Colloquium last week or so. One reads in Caicedo's
abstract:

"We show that fine structural inner models for mild large cardinal
hypotheses but below a Woodin cardinal admit forcing extensions where
bounded forcing axioms hold and the reals are projectively
well-ordered."

If "the reals are projectively well-ordered", is that interpretable as
"the reals are well-ordered" or "the normal ordering of the reals is a
well-ordering on the positive reals"?

I'm glad you wrote, it gave an excuse to ask you.

The normal ordering is a _total_ ordering of the reals. The choice
function of a well-ordering, as you describe, only returns a
real-number without any idea of what it _is_, except for being a real
number.

What do you think about well-ordering the non-negative reals and the
normal ordering as a candidate well-ordering? The provably existent
well-ordering constraints for the reals are semantically vague. In
what ways can you conceive that those definitions can be bent, and not
broken, and allow the normal ordering to be a well-ordering?

Regards,

Ross Finlayson 

--
"On further review of the metric system, I mass 85 kilos."

Relevant Pages

  • Re: Review of Mueckenheims book.
    ... If your understanding concerns a definable well ordering of the reals, ... Yet one cannot prove in ZFC that s ... > contains any definabel member as one cannot prove in ZFC that there is ... > a definable well-ordering of the set of all real numbers. ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... that every set has a well-ordering. ... So, it is known the set of reals, where the collection of all real ... some sequence of reals with the normal ordering. ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... that every set has a well-ordering. ... So, it is known the set of reals, where the collection of all real ... some sequence of reals with the normal ordering. ...
    (sci.logic)
  • Re: Cantor Confusion
    ... >> of the reals is consistent with ZFC. ... It would give a well-ordering. ... Every non empty set of a well-ordering has to have a first element. ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... But it is a theorem that, working in ZFC, you cannot ... > reals. ... Feferman's result about well-ordering the reals. ... The normal ordering is a _total_ ordering of the reals. ...
    (sci.logic)