The Axiom of Constructibility is possibly the most profound and far-reaching idea in all of mathematics. The axiom is very easy to state:

V = L

Unfortunately I never studied it formally, and a Web search has failed to come up with anything that enables me to understand it fully, so here is an informal presentation.

Set theory is the foundation of mathematics, and the foundation of set theory is the Zermelo-Fraenkel axioms (ZF). From this handful of principles ever larger sets can be built up, in spiralling infinite powers. At the heart of this cone is a line of special sets called cardinals, which are the infinite equivalents of numbers (or sizes, if you prefer).

The sets that you can build, starting with nothing and working your way up using the ZF axioms, are called constructible sets.

The open question is: are there any sets that are not constructible? Can you have them? Is their existence consistent with ZF?

The answer is yes; it is independent of ZF whether or not there are sets that are not axiomatically constructible. You can put others in, you can leave others out, both are consistent. The universe of all sets is denoted V. The universe of all constructible sets is denoted L. Trivially, L <= V. The Axiom of Constructibility is that V = L, and it is usually just called V = L in the literature.

It is an immensely powerful, constraining principle. Among the things it proves are another strong principle, called Jensen's diamond (that's written as an actual diamond, not the word diamond), which I don't know. From that follows GCH, the Generalized Continuum Hypothesis, which states that there are no sets of intermediate size between an infinite set and the set of all its subsets. (See neil's write-up under the continuum problem for an excellent and more detailed explanation.)

GCH implies the Axiom of Choice, which is equivalent to Zorn's Lemma and the Well-Ordering Principle, which are used constantly in much of mathematics, but are also provably independent of the usual ZF foundation. So all in all, V = L is an utterly sweeping claim about the whole universe of mathematics. For this reason, some mathematicians embrace it, some reject it.

Log in or register to write something here or to contact authors.