L Has A Reflection Principle
Proving that the axiom of separation, axiom of replacement, and axiom of choice hold in L requires (at least as shown above) the use of a reflection principle for L. Here we describe such a principle.
By mathematical induction on n<ω, we can use ZF in V to prove that for any ordinal α, there is an ordinal β>α such that for any sentence P(z1,...,zk) with z1,...,zk in Lβ and containing fewer than n symbols (counting a constant symbol for an element of Lβ as one symbol) we get that P(z1,...,zk) holds in Lβ if and only if it holds in L.
Read more about this topic: Constructible Universe
Famous quotes containing the words reflection and/or principle:
“Women generally should be taught that the rough life men must needs lead, in order to be healthy, useful and manly men, would preclude the possibility of a great degree of physical perfection, especially in color. It is not a bad reflection to know that in all probability the human animal has endowments enough without aspiring to be the beauty of all creation as well as the ruler.”
—Caroline Nichols Churchill (1833–?)
“I do not mean to exclude altogether the idea of patriotism. I know it exists, and I know it has done much in the present contest. But I will venture to assert, that a great and lasting war can never be supported on this principle alone. It must be aided by a prospect of interest, or some reward.”
—George Washington (1732–1799)