[FOM] What can't be forced?

[joeshipman@aol.com]

Well, the last two examples are not "known to be (relatively) 
consistent (with ZFC)", though the last example is at least relatively 
consistent with ZF.

"Not Con(ZF)" is indeed known to be relatively consistent with ZFC, but 
it would be nice to have an example which was in some way essentially 
different from this.

If we cannot do better than this, I next ask whether there are 
statements which do not hold in M(G) for any generic set G, which do 
not imply Con(ZF) and do not imply ~Con(ZF), and which are interesting 
enough that we might conceivably care whether they are relatively 
consistent with ZFC.

(I am looking for something we KNOW forcing can't help with, which we 
still want to know the consistency status of -- arithmetical statements 
are the best-known realm where forcing doesn't help us, but I'm not 
aware of any non-contrived arithmetical statements whose consistency we 
care about, which we already know not to imply Con(ZF).)

-- JS

-----Original Message-----
From: Robert Lubarsky <robert.lubarsky at comcast.net>
To: 'Foundations of Mathematics' <fom at cs.nyu.edu>
Sent: Tue, 22 May 2007 7:29 am
Subject: Re: [FOM] What can't be forced?










> A more precise question to ask is "if we start with the minimal
> countable transitive model M, which consists of L(alpha) for the 
first
> alpha where this gives a model of ZFC, is there any statement known 
to
> be consistent which does not hold in M(G) for any generic set G"?

Here are some examples, perhaps none of which will satisfy you:
not Con(ZF)
large cardinal axioms
not AC -- To get the failure of Choice of various kinds, typically you 
take
an inner model of a forcing extension. A forcing extension of a model of
Choice will always satisfy Choice.

Bob Lubarsky


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