[FOM] An argument for V = L
Harvey Friedman
hmflogic at gmail.com
Fri Aug 29 18:04:23 EDT 2014
Tim Chow http://www.cs.nyu.edu/pipermail/fom/2014-August/018132.html wrote
Here's an argument for V = L, from the point of view of the "ordinary
> mathematician," that I haven't seen articulated in quite this way before.
I subscribe to this general line of argument, and have given it before
many times on the FOM and elsewhere. At least in general outline. The
essence of the point, which Tim makes in his own very clear and well
written way, is that there is a certain kind of generality that causes
technical difficulties of a kind that are completely irrelevant to the
mathematical purposes at hand. So rather than the generality being
useful or simplifying the presentation, it opens up a Pandora's box of
technical difficulties that simply do not shed any light on the
intended mathematical investigations. This can go by the name of "set
theoretic generality".
History shows that as set theoretic generality gets processed, there is a
loss of interest, and mathematicians learn to avoid it.
Enter V = L. There are a few ways to bring this into the picture. Some ways
would be expected to appeal to some mathematicians, and not to others; and
other ways vice versa. But there are theorems to the effect that for any of
the obvious ways, you really wind up in totally equivalent places.
I have already dealt with the issue Shipman raised that V = L is difficult
to define mathematically. As soon as one gets interested in f.o.m., one can
easily be comfortable with V = L as "every proper transitive class under
epsilon satisfying ZF is V", or defining L as the least proper transitive
class which, under epsilon, satisfies ZF. One can show that this is highly
robust, in the sense that ZF can be replaced by very weak fragments of ZF.
1. As an axiom. This will appeal to some and not to others. Assuming you
are abandoning an interest in set theoretic generality, there is no
compelling argument that there is a nonconstructible set. Since V = L seems
to solve all of the set theoretic problems of limited rank (say in
V(omega+omega)), just assume it, so we can get back to real mathematics.
2. As a relativization. Again, assuming you are abandoning an interest in
set theoretic generality, ..., L has far more in it than would ever come up
mathematically. So we don't lose any mathematics we are interested in if we
simply restrict to L - and besides, there is no compelling argument for a
nonconstructible set from the point of view of those abandoning an interest
in set theoretic generality.
There are well known equivalences between 1 and 2.
Of course, there is a monkey wrench in all this, particularly if the
following Thesis is verified - and we are not quite there yet.
THESIS. Corresponding to every interesting level in the interpretation
hierarchy referred to above, there is a Pi01 sentence of clear mathematical
interest and simplicity. I.e., which is demonstrably equivalent to the
consistency of formal systems corresponding to that level, with the
equivalence proved in EFA (or even less). There are corresponding
formulations in terms of interpretations and conservative extensions.
Then what?
Harvey Friedman
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