Axiom of constructibility
The Konstruierbarkeitsaxiom is going back to Kurt Godel statement of set theory, which is a possible extension of the Zermelo -Fraenkel set theory ZFC. It states that all sets are constructible ( in a specifiable sense), and is usually abbreviated by the equation. This statement can not be derived from ZFC, but one can show that the additional assumption of its correctness can not lead to contradictions that could not have come alone by ZFC to pass. In a lot of the universe, which satisfies ZF and the Konstruierbarkeitsaxiom automatically apply the axiom of choice and the generalized continuum hypothesis, how could show Gödel.
The basic idea for Konstruierbarkeitsaxiom is to make the amount of the universe as small as possible. For this purpose you describe construction processes by so-called fundamental operations and lastly, calls can be in this way already construct all quantities.
Classes as functions
In order to formulate subsequent executions easier, we extend in a first step some known definitions and notation for functions on arbitrary classes:
- Is the class of all, for which there is one with, and is called the domain of.
- Is the class of all, for which there is a with, and is called the range of values .
If a function, we obtain the usual functions for terms of definition and range of values.
- Was on for a class if the couple is in and there are no other couples with.
Otherwise, is defined as the empty set.
If a function, it is as usual the value of the function at the point where if is from the domain, and the same if. The above definition is much more general, it applies to each class.
Eight fundamental operations
It defines eight operations create two volumes and a third.
- , Which is the pair set with the elements and
- . It stands for the element relation. Thus, the result consists of all pairs in with, whatever.
- , The difference amount.
- Which is the set of all pairs with out. Specifically, is a function, so this is the restriction of this function to the crowd.
- . Here, the domain of.
- . Here is the set of all pairs, is for the in.
- . Here is the set of all triples, is for the in.
- . Here is the set of all triples, is for the in.
Construction of sets
In the following step, the eight fundamental operations into a single on, the class of all ordinals, defined function are summarized. The idea is to consider the expression as a function of, with the numbers 1 through to 8, and to construct it as a means of isomorphism function.
In the class you declare the following order:
() Or
( And ) or
( And and ) or
( And and ).
One can show that this defines a well-founded well-ordering on. Therefore there is exactly one Ordnungsisomorphismus.
Next is the -th component of, if is an ordinal number, and otherwise the empty set. This are functions and defined. It has values in; note to that.
Now one defines a function for all quantities as follows:
Finally, on the means of transfinite induction Define the constructor function:
- Is the. defined to function with for all ordinals
The structural hierarchy and the Konstruierbarkeitsaxiom
Usually refers to the amount of the universe, ie the class of all sets, or short. With one denotes the class of all constructible sets, and it is. The design of the elements by means of the ordinal numbers may be on in a simple manner to define a hierarchy Konstruktible hierarchy of classes and.
The relying entirely here question whether any amount is constructible, ie, whether the Konstruierbarkeitsaxiom so-called true, turns out to be undecidable.
If, in the ZF axioms all quantifiers and that one can read as or by the restricted quantifiers or, so you can prove that even limited to, all ZF axioms. In this sense, is a model of ZF. You have to very carefully distinguish between ZF and the model of ZF, which was constructed by ZF here.
In the model, all quantities can be constructed, that is, it is here that Konstruierbarkeitsaxiom. Therefore, one can on the basis IF the existence of non- constructible sets not be inferred, because the same derivation would also apply in the model. In particular, the assumption is as an additional axiom IF not inconsistent with the assumption that ZF is consistent; one speaks of relative consistency. Using model theory one can also show that not from ZF, not even from derivable.
Other axioms
From the Konstruierbarkeitsaxiom some more in ZF alone is not provable statements can be derived, then these are also relatively consistent.
The axiom of choice
For every constructible set with an ordinal number; be it with the least ordinal.
Set. Then one can show that a function is for all non-empty.
Thus, in ZF is valid under the additional assumption of Konstruierbarkeitsaxioms the axiom of choice; more, there is even a universal choice function, namely the above. It just writes.
The axiom of choice AC thus proves to be relatively consistent. In a lot of universe with Konstruierbarkeitsaxiom the axiom of choice is not necessary, because it can be derived.
The generalized continuum hypothesis
Gödel has also shown that ( GCH ) is valid in the generalized continuum hypothesis. In ZF can therefore be concluded from the Konstruierbarkeitsaxiom on GCH, short. It is plausible that one should have the validity of the generalized continuum hypothesis as few volumes in the universe lot, because between the cardinality of an infinite set of cardinality and its power set, it is supposed to be no further widths. This was Gödel's original motivation for the study of constructability.