[FOM] Thoughts on CH 2

joeshipman at aol.com joeshipman at aol.com
Sat Aug 16 02:00:01 EDT 2014

"No detailed structure theory" is such a cop-out.

They simply don't like the axiom and so give a vague standard that is neither described precisely, nor demonstrated for alternative axioms.

Several times on this forum, I have challenged set theorists to explain what is wrong with the real-valued-measure axiom and I have never gotten a response. From the RVM axiom, all kinds of things can be proven about sets of reals, in fact very little of interest is left undecided by this axiom. It also has the consistency strength of a large cardinal axiom (measurable cardinal). What's not to like?

I claim that RVM is intuitively consistent even after the Banach-Tarski paradox is taken into account. If mathematics had happened to develop with this assumed as an axiom, then the Banach-Tarski paradox might have been interpreted to mean "space is not isotropic" rather than "space is not infinitely divisible", and Ulam's result could have been regarded as settling Hilbert's First Problem, and Godel's results on L could have been interpreted as saying that ZFC was in fact consistent (because the weak inaccessibles RVM provides give an actual model when you take L that far up), and Cohen's results about CH could have been regarded as results about how badly RVM could fail that supported its necessity, and Solovay's results could have been given greater importance and regarded as justifying large cardinals up to measurables. Further large cardinals would be investigated via Product Measure Extension Axioms.

On some other planet this path might have been taken and their mathematicians might regard our bafflement over CH as a silly prejudice against real-valued measures.

Now I could be all wrong about this, but to argue that I am wrong you have to do a lot better than "no detailed structure theory" and say why the consequences of RVM are undesirable. This is a tough row to hoe, if you don't have an alternative axiom of comparable simplicity, intuitiveness, and power, and in fact I doubt you can even achieve two out of those three.

-- JS

-----Original Message-----
From: Harvey Friedman <hmflogic at gmail.com>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Fri, Aug 15, 2014 8:25 pm
Subject: [FOM] Thoughts on CH 2

It is my impression that just about all of the current approaches to CH have at their basis, the following flawed "proof" of not CH. 

Let M be a countable transitive model of ZFC. Then we can extend M in the usual way by forcing to a countable transitive model of ZFC M' with the same ordinals such that CH is false. 

In particular, if CH holds in M, then we can add omega_2 mutually Cohen reals and get c = omega_2. This is already due to Cohen. 

So, the "argument" goes, if we want to MAXIMIZE then we must essentially have omega_2 reals and not CH. 

There are a number of flaws in this argument, not the least of which is that "yes, but we can then make another generic extension, adding no new reals, but collapsing omega_2 to omega_1, thereby restoring CH". 

There are other flaws in this argument. 

1. Whereas countable transitive models are obviously "extrinsically" important, they are not at least prima facie "intrinsically" fundamental. On the face of it, having a countable universe of sets must be only some sort of technical device or perhaps some sort of approximation to reality. 

2. Whereas forcing is obviously "extrinsically" important, it is not at least prima facie "intrinsically" fundamental. It is, prima facie, a technical device. 

But the biggest problem that needs to be first addressed is that we can extend to violate CH, and then extend again to restore CH. 

The natural move is of course to declare one of these kinds of extensions illegal. I.e., the second kind. But with what rationale?

An attempted rationale would be the following. Since equinumerosity is to fundamental to set theory, we only want to consider extensions that preserve equinumerosity. I.e., two sets in a model are required to be equinumerous in any legal extension if and only if they are equinumerous in the ground model. 

Now the argument goes like this. Since every countable transitive model has a legal extension with not CH, and every legal extension of every countable transitive model with not CH must have not CH, THEREFORE CH is true according to MAXIMIZE.

So we are using a principle which can be informally stated as follows.

DEFINITION. A universe is a model of ZFC. One universe strongly extends another if and only if it extends it in the usual sense and also preserves equinumerosity.

PRINCIPLE A. Let phi be a sentence. Suppose every universe has an imaginary strong extension satisfying phi. Furthermore, suppose any strong extension of any universe satisfying phi must satisfy phi. Then phi is true in this universe. 

I am under the impression that this sort of fairly simplistic thing has been intensively investigated treating "imaginary universes"  in terms of set forcing and class forcing, and "strong extensions" in terms of ccc forcing extensions, or more generally cardinal preserving set and class forcing extensions. I can see that perhaps some phi chosen to be connected with "Jensen coding up the universe by a real" causes real problems with this approach.  THere are obvious amalgamation issues. 

Assuming that this kind of idea has been dead and buried, I would like to offer some sort of possible partial resurrection as follows. 

Firstly, enumerate a practical FINITE catalog of important, natural, and famous set theoretic conjectures phi_1,...,phi_k, which of course icludes CH. 

Now state Principle A ONLY for phi_1,...,phi_k. What happens? Maybe in practice (in this sense) we can make use of Principle A? Maybe if we restrict to phi of a certain interesting kind (which would include uninteresting cases as usual), Principle A can be salvaged? 

If Principle A can be made to work even in this limited way, using commonplace technical formulations, then I think that issues 1,2 above can be satisfactorily handled foundationally.

Now for another unrelated point. Consider 

*) there is a countably additive probability measure on all sets of reals where the measure of points is 0.

This is well known to imply not CH. Furthermore, it is directly related to a mathematically interesting and well known concept - countably additive probability measures. I would assume that "all" set theorists reject this solution to CH. A challenge for CH research is to explain why proposals like *) and perhaps a workable form of Principle A above are far less satisfactory than current CH research. 

The usual answer I think is that there is no detailed structure theory associated with *). But maybe something interesting structurally can be done with *). 

Harvey Friedman


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