Re: Uncountable sets in CZF?

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 08/18/04 

Date: Wed, 18 Aug 2004 08:48:32 +0200

k_over_hbarc@yahoo.com (Andrew Usher) writes:

> What additional axiom does CZF have? If it is consistent, it would
> disprove the claim of Jesse Hughes that V>=L necessarily.

You are badly confused.

I know that I didn't respond to your question in
<6e197594.0408121932.665412fc@posting.google.com>, but I was hoping
someone that knows the definition of non-measurable would give it a
go. I'll give it my best nonetheless.

You wrote:

,----[ <6e197594.0408121932.665412fc@posting.google.com> ]
| Does the construction of L require AC to begin with?
|
| To go back to a previous example, AC -> there is a non-measurable
| subset of R.
|
| If V=L -> AC, then L must contain such a set.
|
| Clearly, adding more sets to V (and making AC false) can't eliminate
| anything, so there must be a non-measurable regardless of AC, if L is
| contained in V, right?
`----

I don't know diddly about the definitions, but I imagine that the
definition of non-measurable includes some quantifiers. If we enlarge
the universe of discourse, then a set which was previously
non-measurable may well become measurable. Someone that knows a
definition or two can confirm or deny this, but I'd reckon that a set
is not measurable/non-measurable all by its lonesome, but only in some
context, namely the universe.

In any case, it is just painfully obvious that V >= L. L is the
collection of all sets which provably exist. Any model of ZF must
contain every set which provably exists (well, perhaps with some of
these sets identified). Let's call a model M of ZF /good/ if,
whenever M |= s = t for any terms s and t, then also ZF |- s = t.
Some model theorist can tell me what the right terminology is, but
"good" will do for now. Clearly, if M is good, then L <= M. That is,
then L is a submodel of M.

V is clearly a good model of ZF. Therefore L is a submodel (not
necessarily proper) of V.

This is more detail than ought to be necessary. Every constructible
set is a *set*. Therefore, L is contained in V. Duh. 

-- 
"But he himself was not to blame for his vices. They grew out of a personal
defect in his mother. She did her best in the way of flogging him while an
infant... but, poor woman! she had the misfortune to be left-handed, and a
child flogged left-handedly had better be left unflogged." -- E.A. Poe

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