FOM: V=L Conjecture
Harvey Friedman
friedman at math.ohio-state.edu
Sun May 14 00:50:23 EDT 2000
I came up with a very strong reasonably precise formulation of the thesis
*) all statements made in normal mathematics other than those discovered by
Friedman and a few close associates is provable or refutable in ZFC + V = L.
A previous way I had of formulating this was as follows:
**) every mathematically natural statement about mathematical objects lying
within V(omega + omega) other than those discovered by Friedman and a few
close associates is provable or refutable in ZFC + V = L.
However, the normal mathematician will frequently formulate statements
about arbitrary groups, rings, fields, Banach spaces, etcetera, without
regard to their cardinality. They will do this as long as it simplifies
their real purposes. When this great generality causes difficulties
uncharacteristic of normal mathematical purposes, they will simply cut down
the generality to remove such difficulties.
Nevertheless, despite all this, it would be nice to incorporate arbitrary
objects in our formulation. Here is what I came up with :
***) every mathematical natural statement for the form "for all
mathematical objects, such and such holds" where such and such is set
theoretically bounded, other than those discovered by Friedman and a few
close associates, is provable or refutable in ZFC + V = L.
In this context, "set theoretically bounded" is quite robust, and can be
naturally taken to be a set theoretic formula that is bounded in the power
set operation. It can, however, also be taken to be a bounded formula
(power set operation given no special status), and ***) would remain
unchanged.
My counterexamples to these conjectures involve Borel diagonalization in
the 1980's, work in the Annals of Math 1998, and upcoming work in Boolean
relation theory.
A refutation of ***) could involve some mathematically natural formulation
of "every cardinal is not inaccessible." I have never seen such a
statement.
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