[FOM] obscure technical question concerning ZF without choice

[Harvey Friedman]

On Jan 25, 2009, at 9:57 AM, Thomas Forster wrote:


> If we drop AC$_\omega$ then we can no longer prove that a union of
> countably many countable sets is countable.  Let
>
> 	$C_0$ be the class of countable sets;
> 	$C_{\alpha + 1}$ be the class of countable unions of things
> 		in $C_\alpha$;
> 	$C_\lambda$ is the union of earlier $C_\alpha$.
>
> (Use Scott's trick if you are worried about classes: this isn't a  
> question
> about classes)
>
> Must there be an $\alpha$ such that $C_\alpha$ = C_{\alpha +1}$?
>
> Is this known?  (What i am hoping is that as soon as i have lifted  
> my head
> above the parapet in posting this i will discover a simple solution:  
> it
> often works.)  I do remember many years ago being told that someone  
> called
> `Morris' proved (i think i've got this right) that there could be  
> models
> of ZF-minus-choice in which one could find arbitrarily large things  
> that
> were the size of a power set of a countable union of countable sets --
> which has the same sort of flavour.  I would be grateful to anyone  
> telling
> me more about either of these.

I do remember seeing an unpublished copy of the Morris thesis, whose  
adviser was Jerry Keisler. I haven't seen a published version, and I  
don't know if one exists. Morris, if I recall, left academia right  
after getting his PhD. at UW, Madison, around 1970 or so.

If I recall, for background, Levy showed how to force the reals to be  
a countable union of countable sets. I think Morris showed how to  
force that for all ordinals alpha, there is a map from some countable  
union of countable sets onto alpha. Such a transitive forcing  
extension cannot have an ordinal preserving extension of ZFC.

With regard to your technical question, first consider a related  
problem. Define

W(0) = the set of all countable unions of countable sets of reals.
W(alpha + 1) = the set of all countable unions of elements of W(alpha).
W(lambda) = the union of the W(beta), beta < lambda.

Show that over any ctm of ZFC, you can force that the above hierarchy  
goes on for a long time; e.g., that cardinals are preserved, and it  
goes on for (at least) omega_1 steps. Perhaps you can get some strong  
result about stopping times for this, with various kinds of cardinal  
preservation. In any case, show that this can go on for at least omega  
steps.

Then generalize, by considering

W(kappa,0) = the set of all countable unions of countable sets of  
elements of the power set of kappa.
W(kappa,beta + 1) = the set of all countable unions of elements of  
W(kappa,beta).
W(kappa,lambda) = the union of the W(alpha,gamma), gamma < lambda.

Assume kappa is an infinite successor cardinal. Force, say, that this  
can go on for at least alpha steps.

Then put these forcings together for all infinite successor cardinals  
kappa. Then your hierarchy will go on forever in that model.

So let's formulate a plan for the W(alpha) hierarchy above. Let M be a  
ctm of ZFC. Let M* be the forcing extension satisfying ZFC, obtained  
from M by adding a Cohen generic subset A of omega_1. We are going to  
define an inner model of M*. Write A = {a_alpha: alpha < omega_1}.

Suppose f is an omega_1 sequence consisting of omega sequences of  
omega sequences of reals. Then we can obtain another f* from f of this  
kind in the following way. We can use a generic map g from omega_1  
into omega sequences from omega_1, obtained from A. We then get f'  
which is an omega_1 sequence consisting of omega sequences of values  
of f. These omega sequences of values of f are collapsed in a  
canonical way to omega sequences of omega sequences of reals.

Thus we obtain a hierarchy f_0, f_1, ..., along omega, where each f_i 
+1 = f_i*. We start with f_0 obtained in a simple way from A.

It is safe to use different parts of A as we go along these f+i's.  
This is easy to do, since A naturally splits into omega_1 disjoint  
pieces.

At limits, we can take an appropriate "union" using a pairing function  
on omega_1.

Do this to obtain f_alpha, alpha < omega_1. The f_alpha +1 should only  
involve the alpha-th disjoint part of A.

Now in M*, we have constructed omega_1 many: omega sequences of omega  
sequences of reals. We convert these to the associated omega sequences  
of countable sets of reals, by taking ranges.

We now define the inner model M' of M*. It is the least model of ZF  
with the same ordinals, such that

i. every sequence of reals that is used in the f's lies in M'.
ii. every omega sequence of countable sets of reals associated above  
to values of the f's, also lies in M'.

This should work: that the W(alpha) hierarchy above goes on for at  
least omega_1 steps in M', where the omega_1 of M' is the same as the  
omega_1 of M.

And this should generalize over any infinite successor cardinal kappa,  
to get at least kappa length W(kappa,alpha) hierarchy.

I leave all details to others.

Harvey Friedman