Constructibility axiom

The axiom of constructibility is a statement of set theory going back to Kurt Gödel , which represents a possible extension of the Zermelo-Fraenkel set theory ZFC. It says that all sets are constructible (in a definable sense) and is usually abbreviated by the equation . This statement cannot be derived from ZFC, but it can be shown that the additional assumption of its correctness cannot lead to contradictions that could not arise through ZFC alone. In a universe of sets that fulfills ZF and the constructability axiom, the axiom of choice and the generalized continuum hypothesis automatically apply , as Gödel was able to show. ${\displaystyle V=L}$

The basic idea behind the axiom of constructibility is to make the universe of sets as small as possible. For this purpose one describes construction processes by so-called fundamental operations and finally demands that all sets can already be constructed in this way.

Classes as functions

In order to be able to formulate the following explanations more easily, in a first step we extend some known definitions and notations for functions to any classes : ${\displaystyle x}$

• ${\displaystyle D(x)}$is the class of everyone for whom there is a with , and is called the domain of . ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle (y,z)\in x}$ ${\displaystyle x}$
• ${\displaystyle W(x)}$is the class of everyone for which there is a with and is called the range of . ${\displaystyle z}$ ${\displaystyle y}$ ${\displaystyle (y,z)\in x}$ ${\displaystyle x}$

If a function is, then one obtains the terms of definition and value range that are usual for functions. ${\displaystyle x}$

• Continue for a class if the pair is in and there are no other pairs with . ${\displaystyle x}$ ${\displaystyle x(y)=z}$ ${\displaystyle (y,z)}$ ${\displaystyle x}$ ${\displaystyle (y,w)\in x}$ ${\displaystyle z\not =w}$

Otherwise it is defined as an empty set . ${\displaystyle x(y)}$ ${\displaystyle \emptyset }$

If a function is, the value of the function is as usual at the point if it is out of the definition range , and the same if . The above definition is much more general, it applies to every class . ${\displaystyle x}$ ${\displaystyle x(y)}$ ${\displaystyle y}$ ${\displaystyle y}$ ${\displaystyle D(x)}$ ${\displaystyle \emptyset }$ ${\displaystyle y\notin D(x)}$ ${\displaystyle x}$

Eight fundamental operations

Eight operations are defined that generate two sets and a third . ${\displaystyle {\mathcal {F}}_{1},\dotsc ,{\mathcal {F}}_{8}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle {\mathcal {F}}_{i}(a,b)}$

• ${\displaystyle {\mathcal {F}}_{1}(a,b)=\{a,b\}}$, that is the pair set with the elements and ${\displaystyle a}$ ${\displaystyle b}$

• ${\displaystyle {\mathcal {F}}_{2}(a,b)=a\cap {\in }}$. Stands for the element relation. The result consists of all pairs in with , regardless of . ${\displaystyle \in }$ ${\displaystyle (x,y)}$ ${\displaystyle a}$ ${\displaystyle x\in y}$ ${\displaystyle b}$

• ${\displaystyle {\mathcal {F}}_{3}(a,b)=a\setminus b}$, the difference amount .

• ${\displaystyle {\mathcal {F}}_{4}(a,b)=a|_{b}}$, that's the set of all the couples out with . If a function is special , this is the limitation of this function to the set . ${\displaystyle (x,y)}$ ${\displaystyle a}$ ${\displaystyle x\in b}$ ${\displaystyle a}$ ${\displaystyle b}$

• ${\displaystyle {\mathcal {F}}_{5}(a,b)=a\cap D(b)}$. It is the domain of . ${\displaystyle D(b)}$ ${\displaystyle b}$

• ${\displaystyle {\mathcal {F}}_{6}(a,b)=a\cap b^{-1}}$. Where is the set of all pairs for which in lies. ${\displaystyle b^{-1}}$ ${\displaystyle (y,x)}$ ${\displaystyle (x,y)}$ ${\displaystyle b}$

• ${\displaystyle {\mathcal {F}}_{7}(a,b)=a\cap {\rm {cnv}}_{2}(b)}$. Where is the set of all triples for which in lies. ${\displaystyle {\rm {cnv}}_{2}(b)}$ ${\displaystyle (z,x,y)}$ ${\displaystyle (x,y,z)}$ ${\displaystyle b}$

• ${\displaystyle {\mathcal {F}}_{8}(a,b)=a\cap {\rm {cnv}}_{3}(b)}$. Where is the set of all triples for which in lies. ${\displaystyle {\rm {cnv}}_{3}(b)}$ ${\displaystyle (x,z,y)}$ ${\displaystyle (x,y,z)}$ ${\displaystyle b}$

Construction of sets

In the following step, the eight fundamental operations are combined into a single function defined on the class of all ordinal numbers . The idea is to view the expression as a function of , with the numbers running through from 1 to 8, and to construct this as a function of using an isomorphism . ${\displaystyle On}$ ${\displaystyle F}$ ${\displaystyle {\mathcal {F}}_{i}(a,b)}$ ${\displaystyle (a,b,i)}$ ${\displaystyle i}$ ${\displaystyle On\times On\times \{0,\dotsc ,8\}\rightarrow On}$ ${\displaystyle On}$

On the class, explain the following order: ${\displaystyle On\times On\times \{0,\dotsc ,8\}}$ ${\displaystyle (\alpha ,\beta ,m)<(\gamma ,\delta ,n):\Leftrightarrow }$

( ) or ${\displaystyle \max\{\alpha ,\beta \}<\max\{\gamma ,\delta \})}$

( and ) or ${\displaystyle \max\{\alpha ,\beta \}=\max\{\gamma ,\delta \})}$ ${\displaystyle \alpha <\gamma }$

( and and ) or ${\displaystyle \max\{\alpha ,\beta \}=\max\{\gamma ,\delta \})}$ ${\displaystyle \alpha =\gamma }$ ${\displaystyle \beta <\delta }$

( and and ). ${\displaystyle \alpha =\gamma }$ ${\displaystyle \beta =\delta }$ ${\displaystyle m<n}$

One can show that this is a sound well-ordering on defined. Therefore there is exactly one order isomorphism . ${\displaystyle On\times On\times \{0,\dotsc ,8\}}$ ${\displaystyle J\colon On\times On\times \{0,\dotsc ,8\}\rightarrow On}$

Furthermore, let the -th component of if is an ordinal number and the empty set otherwise. This defines functions and . Has values ​​in ; note that . ${\displaystyle K_{j}(x)}$ ${\displaystyle j}$ ${\displaystyle J^{-1}(x)}$ ${\displaystyle x}$ ${\displaystyle K_{1},K_{2}}$ ${\displaystyle K_{3}}$ ${\displaystyle K_{3}}$ ${\displaystyle \{0,\dotsc ,8\}}$ ${\displaystyle \emptyset =0}$

Now we define a function for all sets as follows: ${\displaystyle G}$ ${\displaystyle x}$

${\displaystyle G(x)={\begin{cases}W(x),&{\mbox{falls }}K_{3}(D(x))=0\\{\mathcal {F}}_{i}(x(K_{1}(D(x))),x(K_{2}(D(x)))),&{\mbox{falls }}K_{3}(D(x))=i>0\end{cases}}}$

Finally, means can be transfinite induction of the structure function defining: ${\displaystyle G}$ ${\displaystyle F}$

• ${\displaystyle F}$is the function defined on with for all ordinal numbers . ${\displaystyle On}$ ${\displaystyle F(\alpha )\,=\,G(F|_{\alpha })}$ ${\displaystyle \alpha \in On}$

A set is now called constructible if there is an ordinal with . The first examples constructible sets are , , , , , , , , , ${\displaystyle x}$ ${\displaystyle \alpha }$ ${\displaystyle x=F(\alpha )}$ ${\displaystyle F(0)=0=\emptyset }$ ${\displaystyle F(1)=1=\{0\}}$ ${\displaystyle F(2)=0}$ ${\displaystyle F(3)=0}$ ${\displaystyle F(4)=0}$ ${\displaystyle F(5)=0}$ ${\displaystyle F(6)=0}$ ${\displaystyle F(7)=0}$ ${\displaystyle F(8)=0}$ ${\displaystyle F(9)=2=\{0,1\},\dotsc }$

The constructive hierarchy and the constructibility axiom

Usually one designates with the set universe, that means the class of all sets, or in short . With denotes the class of all constructible sets, and it holds . By constructing the elements of with the help of ordinal numbers, you can easily define a hierarchy, the constructible hierarchy of classes with   and   . ${\displaystyle V}$ ${\displaystyle V=\{x;\,x=x\}}$ ${\displaystyle L}$ ${\displaystyle L\subseteq V}$ ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle L_{\alpha }}$ ${\displaystyle \alpha <\beta \implies L_{\alpha }\subseteq L_{\beta }}$ ${\displaystyle L=\bigcup _{\alpha \in Ord}L_{\alpha }}$

The question that arises here, whether every set can be constructed, i.e. whether the so-called axiom of constructibility applies, proves to be indecisable. ${\displaystyle V=L}$

If one replaces all quantifiers or in the ZF-axioms , which one can read as or , by the restricted quantifiers or , then one can prove that all ZF-axioms also apply then, restricted to . In this sense is a model for ZF. A very careful distinction must be made here between the ZF and the model for ZF, which was constructed using the ZF. ${\displaystyle \forall x}$ ${\displaystyle \exists x}$ ${\displaystyle \forall x\in V}$ ${\displaystyle \exists x\in V}$ ${\displaystyle \forall x\in L}$ ${\displaystyle \exists x\in L}$ ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle L}$

In the model, all sets can be constructed, that is, it is here the axiom of constructibility . Therefore one cannot derive the existence of non-constructible sets on the basis of ZF, because the same derivation would have to apply in the model . In particular, the assumption as an additional axiom to ZF is not contradicting the assumption that ZF is free of contradictions; one speaks of relative consistency. Using model theory , one can also show that it cannot be derived from ZF, not even from . ${\displaystyle L}$ ${\displaystyle V=L}$ ${\displaystyle L}$ ${\displaystyle V=L}$ ${\displaystyle V=L}$ ${\displaystyle ZFC+GCH}$

Further axioms

From the constructability axiom , some further statements that cannot be proven in ZF alone can be derived; these are then also relatively consistent. ${\displaystyle V=L}$

The axiom of choice

For every constructable set there is an ordinal number with ; let it be the smallest ordinal number with . ${\displaystyle x\in L}$ ${\displaystyle \alpha }$ ${\displaystyle x=F(\alpha )}$ ${\displaystyle {\rm {Od}}(x)}$ ${\displaystyle \alpha }$ ${\displaystyle x=F(\alpha )}$

Set . Then one can show that a function is non-empty with for all . ${\displaystyle A:=\{(x,y);x,y\in L,y\in x,\forall z\in x:{\rm {Od}}(y)\leq \mathrm {Od} (z)\}}$ ${\displaystyle A}$ ${\displaystyle A(x)\in x}$ ${\displaystyle x\in L}$

The axiom of choice thus applies in ZF, assuming the axiom of constructibility; even more, there is even a universal selection function, namely the above . One writes briefly . ${\displaystyle A}$ ${\displaystyle V=L\rightarrow AC}$

The axiom of choice AC thus proves to be relatively consistent. In a set universe with an axiom of constructibility, the axiom of choice is dispensable, because it can be derived.

The generalized continuum hypothesis

Godel has also shown that the generalized continuum hypothesis (GCH) holds true. In ZF, the constructability axiom can be used to infer GCH, in short . It is plausible that for the generalized continuum hypothesis to be valid, one should have as few sets as possible in the set universe, because there should be no further thicknesses between the thickness of an infinite set and the thickness of its power set . This was Godel's original motivation for investigating constructibility. ${\displaystyle L}$ ${\displaystyle V=L\rightarrow GCH}$

The Suslin Hypothesis

The Suslin hypothesis is wrong, as Ronald Jensen was able to show in 1968. ${\displaystyle L}$

literature

• Kurt Gödel : The Consistency of the Axiom of Choice and of the generalized Continuum-Hypothesis with the Axioms of Set Theory (= Annals of Mathematics Studies. Vol. 3). Princeton University Press, Princeton NJ et al. 1940.
• Ronald Jensen : Souslin's hypothesis is incompatible with V = L. In: Notices of the American Mathematical Society. Vol. 15, 1968, ISSN  0002-9920 , p. 935.
• Gaisi Takeuti, Wilson M. Zaring: Introduction to Axiomatic Set Theory (= Graduate Texts in Mathematics. Vol. 1, ZDB -ID 2156806-6 ). Springer, New York NY et al. 1971.