Gödel constructive set

Gödel constructible set, constructible set

A set arising in the process of constructing sets described below. Let be a set and let . Consider the first-order language containing one -place predicate symbol denoting the relation , and individual constants, denoting the elements of the set (for each its constant is ). The statement "the formula f of the language LR, X is valid in the model M=X, R" is written:

A set is called definable in the model (or -definable) if there exists a formula of with one free variable such that

Let denote the set of all -definable sets. To each ordinal number is associated the set defined recursively by the relation

where is the membership relation restricted to the set . Hence it follows that

A set is called constructible if there exists an ordinal number such that . The class of all constructible sets is denoted by . In 1938, K. Gödel defined and introduced the following axiom of constructibility: Every set is constructible. On the basis of the axioms of , he proved that in all axioms of hold and also the axiom of constructibility, and that the axiom of choice and the generalized continuum hypothesis ( "for every ordinal number a one has 2a=a+ 1" ) follow in from the axiom of constructibility.

The class can also be characterized as the smallest class that is a model of and contains all the ordinal numbers; there are other ways of defining (see [2]–[4]). The relation can be expressed by a formula in the language , which is moreover of a simple syntactic structure (a so-called -formula, cf. ).

Some results relating to constructible sets. The set of constructible real numbers (cf. Constructive real number), that is, the set where is the set of all real numbers, that is, sequences of zeros and ones, is a -set (see [5]). It has been shown that the axiom of constructibility implies the existence of a Lebesgue non-measurable set of real numbers of type (see [6]), the negation of the Suslin hypothesis and the non-existence of measurable cardinal numbers (see [2]).

References

[1a]	K. Gödel, "The consistency of the axiom of choice and of the generalized coninuum hypothesis" Proc. Nat. Acad. Sci. USA , 24 (1938) pp. 556–557
[1b]	K. Gödel, "Consistency proof for the generalized coninuum hypothesis" Proc. Nat. Acad. Sci. USA , 25 (1939) pp. 220–224
[2]	T.J. Jech, "Lectures in set theory: with particular emphasis on the method of forcing" , Lect. notes in math. , 217 , Springer (1971)
[3]	A. Mostowski, "Constructible sets with applications" , North-Holland (1969)
[4]	C. Karp, "A proof of the relative consistency of the continuum hypothesis" J. Crossley (ed.) , Sets, models and recursion theory , North-Holland (1967) pp. 1–32
[5]	J.W. Addison, "Some consequences of the axiom of constructibility" Fund. Math. , 46 (1959) pp. 337–357
[6]	P.S. Novikov, "On the non-contradictability of certain propositions of descriptive set theory" Trudy Mat. Inst. Steklov. , 38 (1951) pp. 279–316 (In Russian)
[7]	U. Felgner, "Models of -set theory" , Springer (1971)

Comments

Concerning (the notation) see Descriptive set theory.

As a consequence of Gödel's findings, if the axioms of are non-contradictory, they remain so after addition of the axiom of choice and the generalized continuum hypothesis. This was the first relative consistency result for the theory of any importance, to be surpassed only after a quarter of a century in 1963 by P. Cohen's forcing method. By forcing it is known that cannot prove the axiom of constructibility (unless it is contradictory). Most set theorists think there are no sufficient reasons to believe it to be true. Nevertheless, is an important subclass of the set-theoretic universe well worth investigating.

New results can be found in [a1], which is also a good introduction to constructibility. Reference [a2] contains (most of) the material touched upon in the main article.

References

[a1]	K.J. Devlin, "Constructibility" , Springer (1984)
[a2]	T.J. Jech, "Set theory" , Acad. Press (1978) pp. 523ff (Translated from German)
[a3]	K. Kunen, "Set theory, an introduction to independence proofs" , North-Holland (1980)
[a4]	K. Gödel, "The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory" , Princeton Univ. Press (1940)
[a5]	K. Devlin, "Constructibility" J. Barwise (ed.) , Handbook of mathematical logic , North-Holland (1977) pp. 453–490