[FOM] set theory, V = L, CH

[Joe Shipman]

One thing to like about this formulation is that you can take out the word "proper" and you get the axiom V=M, where M is the class of "strongly constructible sets" defined by Cohen.

If this class forms a set then M is Cohen's "minimal model" but if it doesn't then you settle all large cardinal questions (negatively). It seems to me that if you are trying to justify V=L by saying "only sets that must exist do exist" there is no reason not to apply this philosophy "vertically" as well as "horizontally".

-- JS

Sent from my iPhone

> On Aug 24, 2014, at 12:33 AM, Harvey Friedman <hmflogic at gmail.com> wrote:
> 
> Joe Shipman wrote
> 
> I disagree that ordinary mathematicians would regard V=L as a good axiom, because it cannot even be STATED without bringing in much more set theory than most mathematicians are comfortable with.
> 
> Can you provide an alternative to V=L, consistent with it, that has similar consequences at the level of statements of descriptive set theory, but which is recognizably "mathematical"?
> *****************
> 
> In NBG, formulate V = L as follows. If (A,epsilon) satisfies ZF, where A is a transitive proper class, then V = A. For mathematicians sensitive to the foundational issues, they are going to be comfortable with ZF, and so this will work fine for them. Any alternative is far worse from their point of view. Of course, as perfect Pi01 develops, this becomes a totally new situation. 
> 
> **********
> 
> Monroe Eskew has acted productively on my suggestion that he present some examples of interaction between higher set theory and other areas of mathematics. He came up with some items from functional analysis. This area is put into a better category than general topology by the mathematical community. I will do a little investigation on Eskew's letter and hopefully get back to the FOM about it soon. One thing is already obvious - mathematicians in many areas of math are going to have zero interest in such things, and the issue is this: do they have zero interest in such things simply because it is not their area, or because they are aware of a heavy set theoretic component, or because they instinctively feel a heavy set theoretic component?
> 
> Harvey Friedman
> 
>   
> 
> 
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