Wikipedia / Constructible universe
Wikipedia / Constructible universe
In set theory, the constructible universe (or Gödel's constructible universe is a particular class of sets is the union of the constructible hierarchy. It was introduced by Kurt Gödel in 1938, who proved that the constructible universe is an inner model of ZF set theory, and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.
Definition
can be thought of as being built in "stages" resembling the construction of the von Neumann universe. The stages are indexed by ordinals. In von Neumann's universe, at a successor to be the set of all, one uses only those subsets of the previous stage that are:
definable by a formula in the formal language of set theory,
with parameters from the previous stage, and
with the quantifiers interpreted to range over the previous stage.
By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resulting sets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory and contained in any such model.
Now define the Def operator by
is defined by transfinite recursion as follows:
.
.
is a limit ordinal.
Here Ord denotes the class of all ordinals.
, then
.
, which is a subset of the power set. Consequently, this is a tower of nested transitive sets itself is a proper class.
itself is the "constructible universe". The "axiom of constructibility.
Additional facts about the sets Lα
,
.
: their elements are exactly the hereditarily finite sets. Equality beyond this point does not hold. Even in models of ZFC is inaccessible (see hereditarily countable set#Generalizations).
is an infinite ordinal then there is a bijection, and the bijection is constructible. So these sets are equinumerous in any transitive model of set theory that includes them.
formulas (with respect to the Lévy hierarchy, i.e., formulas of set theory containing only bounded quantifiers and its elements.
under a collection of nine explicit functions, similar to Gödel operations. This definition makes no reference to definability.
All arithmetical).
All hyperarithmetical stands for the Church–Kleene ordinal is hyperarithmetical.
L is a standard inner model of ZFC
is a standard transitive model is a transitive class and the interpretation uses the real element relationship, so it is well-founded is an inner model is a model of ZFC, which means that it satisfies the following axioms:
Axiom of regularity are disjoint sets.
because we are using the same element relation and no new sets were added.
Axiom of extensionality: Two sets are the same if they have the same elements.
).
Axiom of empty set: {} is a set.
.
Axiom of pairing is a set.
.
.
Axiom of separation is a set.
), the latter is called the "reflection principle.
Axiom of replacement is a set.
is.
One can show that there is a definable well-ordering of
be a model of ZF, i.e. we do not.
L is absolute and minimal
.
is the smallest class containing all the ordinals that is an inner model is the intersection of all such classes.
If there is a set. This set is called the minimal model of ZFC. Using the downward Löwenheim–Skolem theorem, one can show that the minimal model (if it exists) is a countable set.
Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets that are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, they do not use the normal element relation and they are not well founded.
does not hold in any other standard model of ZF.
L and large cardinals
. Hence initial ordinals. Regular ordinals. Weak limit cardinals because the generalized continuum hypothesis. Weakly inaccessible cardinals become strongly inaccessible. Weakly Mahlo cardinals become strongly Mahlo. And more generally, any large cardinal property weaker than 0# (see the list of large cardinal properties.
which they also possess. For example, measurable cardinals.
, then there is a closed unbounded class of ordinals that are order-indiscernible. Furthermore, any strictly increasing class function from this class of order-indiscernibles to itself can be extended in a unique way to an elementary embedding. a nice structure of repeating segments.
L can be well-ordered
. Some of these involve the L, which was first described by Ronald Bjorn Jensen could be well-ordered using only the definition given above.
.
. If one discounts (for the moment) the parameters, the formulas can be given a standard Gödel numbering.
), etc. This is called the reverse lexicographic ordering.
.
. And this formula gives the same truth value.
) is equivalent to the axiom of global choice, which is more powerful than the ordinary axiom of choice because it also covers proper classes of non-empty sets.
L has a reflection principle
Proving that the axiom of separation, axiom of replacement, and axiom of choice requires (at least as shown above) the use of a reflection principle. Here we describe such a principle.
.
The generalized continuum hypothesis holds in L
, so . By the downward Löwenheim–Skolem theorem and Mostowski collapse, it is also true in .
. But this is precisely the generalized continuum hypothesis.
Constructible sets are definable from the ordinals
as parameters.
that satisfies the formula:
is short for:
and that contains parameters only for ordinals.
Relative constructibility
.
and all the ordinals.
can be defined by transfinite recursion as follows:
as an element, i.e. the transitive closure.
.
.
.
, the smallest model that contains all the real numbers, which is used extensively in modern descriptive set theory.
.
is a set of ordinals.
itself.
See also
Notes
References