# [FOM] V = L mathematically/corrected

Harvey Friedman friedman at math.ohio-state.edu
Mon Feb 13 09:59:31 EST 2006

```I should have written (TC({A}),epsilon) instead of (TC(A),epsilon). Also I
would rather use "projection" instead of "cross section".

We also make the axiom less intensively set theoretic, not relying on von
Neumann ordinals.

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The crucial definition is that of an inductive relation on a well ordered
set.

Let < be a well ordering. (Minor point: since < is irreflexive, all nonempty
well orderings have at least two field elements). We say that R containedin
fld(<)^2k is strictly dominating if and only if for all x,y in fld(<)^k,
R(x,y) implies max(x) < max(y). For such R, we define R# to be the unique E
containedin fld(<)^k such that E = fld(<)^k\R[E].

We say that R containedin fld(<)^k is order invariant if and only if for all
x,y in fld(<)^k of the same order type, R(x) iff R(y).

Finally, we say that S is an inductive relation on a well ordering < if and
only if S = R# for some strictly dominating order invariant R contained in
some fld(<)^k.

THEOREM 1. The following are provably equivalent in ZF.

i. Every binary (multivariate) relation is a projection of some inductive
relation on some well ordering.
ii. Every binary (multivariate) relation is isomorphic to a projection of
some inductive relation on some well ordering.
iii. V = L.

THEOREM 2. The following are provably equivalent in ZF.

i. {(a,b): a,b in TC({x}) and a epsilon b} is isomorphic to the projection
of some inductive relation on some well ordering.
ii. x is constructible.

Harvey Friedman

```