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''Gödel constructible set, constructible set''
 
''Gödel constructible set, constructible set''
  
A set arising in the process of constructing sets described below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g0445201.png" /> be a set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g0445202.png" />. Consider the first-order language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g0445203.png" /> containing one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g0445204.png" />-place predicate symbol denoting the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g0445205.png" />, and individual constants, denoting the elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g0445206.png" /> (for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g0445207.png" /> its constant is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g0445208.png" />). The statement "the formula f of the language LR, X is valid in the model M=X, Ris written:
+
A set that arises in the process of constructing sets described below. Let $ X $ be a set and $ R \subseteq X \times X $ a relation on $ X $. Then consider the first-order language $ L(R,X) $ containing (i) a binary predicate symbol $ \underline{R} $ denoting the relation $ R $ and (ii) individual constant symbols denoting the elements of $ X $ (for each $ x \in X $, its corresponding constant symbol is $ \underline{x} $). The statement “the formula $ \phi $ of the language $ L(R,X) $ is valid in the model $ M = (X,R) $” is written as
 +
$$
 +
M \models \phi.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g0445209.png" /></td> </tr></table>
+
A set $ Y \subseteq X $ is called “definable” in the model $ M = (X,R) $ (or $ M $-definable) if and only if there exists a formula $ \phi(v) $ of $ L(R,X) $ with one free variable $ v $ such that
 +
$$
 +
\forall x \in X: \quad x \in Y \iff M \models \phi(x).
 +
$$
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452010.png" /> is called definable in the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452011.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452012.png" />-definable) if there exists a formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452014.png" /> with one free variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452015.png" /> such that
+
Let $ \operatorname{Def}(M) $ denote the set of all $ M $-definable sets. To each ordinal $ \alpha $ is associated the set $ L_{\alpha} $ that is recursively defined by the relation
 +
$$
 +
L_{\alpha} = \bigcup_{\beta < \alpha} \operatorname{Def} \left( L_{\beta},\in \!\! |_{L_{\beta}} \right),
 +
$$
 +
where $ \in \!\! |_{L_{\beta}} $ denotes the membership relation restricted to $ L_{\beta} $. Hence, it follows that
 +
\begin{align}
 +
L_{0} & = \varnothing, \\
 +
L_{1} & = \{ \varnothing \}, \\
 +
L_{2} & = \{ \varnothing,\{ \varnothing \} \}, \\
 +
& \vdots \\
 +
L_{\omega} & = \bigcup_{n < \omega} L_{n}, \\
 +
& \vdots
 +
\end{align}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452016.png" /></td> </tr></table>
+
A set $ z $ is called “constructible” if and only if there exists an ordinal $ \alpha $ such that $ z \in L_{\alpha} $. The class of all constructible sets is denoted by $ L $. In 1938, Kurt Gödel defined $ L $ and introduced the following axiom of constructibility: Every set is constructible. On the basis of the axioms of $ \mathsf{ZF} $, he proved that in $ L $, all axioms of $ \mathsf{ZF} $ hold as well as the axiom of constructibility, and that the axiom of choice and the generalized continuum hypothesis (“for every ordinal $ \alpha $, one has $ 2^{\aleph_{\alpha}} = \aleph_{\alpha + 1} $”) follow in $ \mathsf{ZF} $ from the axiom of constructibility.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452017.png" /> denote the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452018.png" />-definable sets. To each ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452019.png" /> is associated the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452020.png" /> defined recursively by the relation
+
The class $ L $ can also be characterized as the smallest class that is a model of $ \mathsf{ZF} $ and contains all the ordinals; there are other ways of defining $ L $ (see [[#References|[2]]]–[[#References|[4]]]). The relation $ z \in L_{\alpha} $ can be expressed by a formula in the language of $ \mathsf{ZF} $, which is moreover of a simple syntactic structure (a so-called $ \Delta_{1}^{\mathsf{ZF}} $, cf. ).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452021.png" /></td> </tr></table>
+
Some results relating to constructible sets. The set of constructible real numbers (cf. [[Constructive real number|Constructive Real Number]]), that is, the set $ \mathbb{R} \cap L $ (where $ \mathbb{R} $ is the set of all real numbers, viewed as sequences of $ 0 $’s and $ 1 $’s), is a $ \Sigma_{1}^{2} $-set (see [[#References|[5]]]). It has been shown that the axiom of constructibility implies the existence of a non-Lebesgue-measurable set of real numbers of type $ \Sigma_{1}^{2} $ (see [[#References|[6]]]), the negation of the [[Suslin hypothesis|Suslin Hypothesis]] and the non-existence of measurable cardinals (see [[#References|[2]]]).
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452022.png" /> is the membership relation restricted to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452023.png" />. Hence it follows that
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452024.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452025.png" /></td> </tr></table>
 
 
 
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452026.png" /> is called constructible if there exists an ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452028.png" />. The class of all constructible sets is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452029.png" />. In 1938, K. Gödel defined <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452030.png" /> and introduced the following axiom of constructibility: Every set is constructible. On the basis of the axioms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452031.png" />, he proved that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452032.png" /> all axioms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452033.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452034.png" /> from the axiom of constructibility.
 
 
 
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452035.png" /> can also be characterized as the smallest class that is a model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452036.png" /> and contains all the ordinal numbers; there are other ways of defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452037.png" /> (see [[#References|[2]]]–[[#References|[4]]]). The relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452038.png" /> can be expressed by a formula in the language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452039.png" />, which is moreover of a simple syntactic structure (a so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452040.png" />-formula, cf. ).
 
 
 
Some results relating to constructible sets. The set of constructible real numbers (cf. [[Constructive real number|Constructive real number]]), that is, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452041.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452042.png" /> is the set of all real numbers, that is, sequences of zeros and ones, is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452043.png" />-set (see [[#References|[5]]]). It has been shown that the axiom of constructibility implies the existence of a Lebesgue non-measurable set of real numbers of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452044.png" /> (see [[#References|[6]]]), the negation of the [[Suslin hypothesis|Suslin hypothesis]] and the non-existence of measurable cardinal numbers (see [[#References|[2]]]).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> K. Gödel,   "Consistency proof for the generalized coninuum hypothesis"  ''Proc. Nat. Acad. Sci. USA'' , '''25''' (1939) pp. 220–224</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> T.J. Jech,   "Lectures in set theory: with particular emphasis on the method of forcing" , ''Lect. notes in math.'' , '''217''' , Springer (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Mostowski,   "Constructible sets with applications" , North-Holland (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.W. Addison,   "Some consequences of the axiom of constructibility"  ''Fund. Math.'' , '''46''' (1959) pp. 337–357</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> U. Felgner,   "Models of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452045.png" />-set theory" , Springer (1971)</TD></TR></table>
+
<table>
 
+
<TR><TD valign="top">[1a]</TD><TD valign="top"> K. Gödel, “The consistency of the axiom of choice and of the generalized continuum hypothesis”, ''Proc. Nat. Acad. Sci. USA'', '''24''' (1938), pp. 556–557. </TD></TR>
 
+
<TR><TD valign="top">[1b]</TD><TD valign="top"> K. Gödel, “Consistency proof for the generalized continuum hypothesis”, ''Proc. Nat. Acad. Sci. USA'', '''25''' (1939), pp. 220–224. </TD></TR>
 +
<TR><TD valign="top">[2]</TD><TD valign="top"> T.J. Jech, “Lectures in set theory: with particular emphasis on the method of forcing”, ''Lect. Notes in Math.'', '''217''', Springer (1971). </TD></TR>
 +
<TR><TD valign="top">[3]</TD><TD valign="top"> A. Mostowski, “Constructible sets with applications”, North-Holland (1969). </TD></TR>
 +
<TR><TD valign="top">[4]</TD><TD valign="top"> 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. </TD></TR>
 +
<TR><TD valign="top">[5]</TD><TD valign="top"> J.W. Addison, “Some consequences of the axiom of constructibility”, ''Fund. Math.'', '''46''' (1959), pp. 337–357. </TD></TR>
 +
<TR><TD valign="top">[6]</TD><TD valign="top"> 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). </TD></TR>
 +
<TR><TD valign="top">[7]</TD><TD valign="top"> U. Felgner, “Models of $ \mathsf{ZF} $-set theory”, Springer (1971). </TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
Concerning (the notation) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452046.png" /> see [[Descriptive set theory|Descriptive set theory]].
+
Concerning (the notation) $ \Sigma_{1}^{2} $, see [[Descriptive set theory|Descriptive Set Theory]].
  
As a consequence of Gödel's findings, if the axioms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452047.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452048.png" /> of any importance, to be surpassed only after a quarter of a century in 1963 by P. Cohen's [[Forcing method|forcing method]]. By forcing it is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452049.png" /> 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044520/g04452050.png" /> is an important subclass of the set-theoretic universe well worth investigating.
+
As a consequence of Gödel’s findings, if the axioms of $ \mathsf{ZF} $ are non-contradictory, then they remain so after adding the axiom of choice and the generalized continuum hypothesis. This was the first relative consistency result of any importance for $ \mathsf{ZF} $, to be surpassed only after a quarter of a century in 1963 by Paul Cohen’s [[Forcing method|method of forcing]]. By forcing, it is known that $ \mathsf{ZF} $ cannot prove the axiom of constructibility (unless it is contradictory). Most set theorists think that there are no sufficient reasons to believe it to be true. Nevertheless, $ L $ is an important subclass of the set-theoretic universe that is well worth investigating.
  
New results can be found in [[#References|[a1]]], which is also a good introduction to constructibility. Reference [[#References|[a2]]] contains (most of) the material touched upon in the main article.
+
New results can be found in [[#References|[a1]]], which is also a good introduction to the concept of constructibility. Reference [[#References|[a2]]] contains (most of) the material touched upon in the main article.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.J. Devlin,   "Constructibility" , Springer (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> T.J. Jech,   "Set theory" , Acad. Press (1978) pp. 523ff (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Kunen,   "Set theory, an introduction to independence proofs" , North-Holland (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> K. Devlin,   "Constructibility"  J. Barwise (ed.) , ''Handbook of mathematical logic'' , North-Holland (1977) pp. 453–490</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> K.J. Devlin, “Constructibility”, Springer (1984). </TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> T.J. Jech, “Set theory”, Acad. Press (1978), pp. 523ff (translated from German). </TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Kunen, “Set theory, an introduction to independence proofs”, North-Holland (1980). </TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> 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). </TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top"> K. Devlin, “Constructibility”, J. Barwise (ed.), ''Handbook of mathematical logic'', North-Holland (1977), pp. 453–490. </TD></TR>
 +
</table>

Revision as of 21:22, 29 January 2016

Gödel constructible set, constructible set

A set that arises in the process of constructing sets described below. Let $ X $ be a set and $ R \subseteq X \times X $ a relation on $ X $. Then consider the first-order language $ L(R,X) $ containing (i) a binary predicate symbol $ \underline{R} $ denoting the relation $ R $ and (ii) individual constant symbols denoting the elements of $ X $ (for each $ x \in X $, its corresponding constant symbol is $ \underline{x} $). The statement “the formula $ \phi $ of the language $ L(R,X) $ is valid in the model $ M = (X,R) $” is written as $$ M \models \phi. $$

A set $ Y \subseteq X $ is called “definable” in the model $ M = (X,R) $ (or $ M $-definable) if and only if there exists a formula $ \phi(v) $ of $ L(R,X) $ with one free variable $ v $ such that $$ \forall x \in X: \quad x \in Y \iff M \models \phi(x). $$

Let $ \operatorname{Def}(M) $ denote the set of all $ M $-definable sets. To each ordinal $ \alpha $ is associated the set $ L_{\alpha} $ that is recursively defined by the relation $$ L_{\alpha} = \bigcup_{\beta < \alpha} \operatorname{Def} \left( L_{\beta},\in \!\! |_{L_{\beta}} \right), $$ where $ \in \!\! |_{L_{\beta}} $ denotes the membership relation restricted to $ L_{\beta} $. Hence, it follows that \begin{align} L_{0} & = \varnothing, \\ L_{1} & = \{ \varnothing \}, \\ L_{2} & = \{ \varnothing,\{ \varnothing \} \}, \\ & \vdots \\ L_{\omega} & = \bigcup_{n < \omega} L_{n}, \\ & \vdots \end{align}

A set $ z $ is called “constructible” if and only if there exists an ordinal $ \alpha $ such that $ z \in L_{\alpha} $. The class of all constructible sets is denoted by $ L $. In 1938, Kurt Gödel defined $ L $ and introduced the following axiom of constructibility: Every set is constructible. On the basis of the axioms of $ \mathsf{ZF} $, he proved that in $ L $, all axioms of $ \mathsf{ZF} $ hold as well as the axiom of constructibility, and that the axiom of choice and the generalized continuum hypothesis (“for every ordinal $ \alpha $, one has $ 2^{\aleph_{\alpha}} = \aleph_{\alpha + 1} $”) follow in $ \mathsf{ZF} $ from the axiom of constructibility.

The class $ L $ can also be characterized as the smallest class that is a model of $ \mathsf{ZF} $ and contains all the ordinals; there are other ways of defining $ L $ (see [2][4]). The relation $ z \in L_{\alpha} $ can be expressed by a formula in the language of $ \mathsf{ZF} $, which is moreover of a simple syntactic structure (a so-called $ \Delta_{1}^{\mathsf{ZF}} $, cf. ).

Some results relating to constructible sets. The set of constructible real numbers (cf. Constructive Real Number), that is, the set $ \mathbb{R} \cap L $ (where $ \mathbb{R} $ is the set of all real numbers, viewed as sequences of $ 0 $’s and $ 1 $’s), is a $ \Sigma_{1}^{2} $-set (see [5]). It has been shown that the axiom of constructibility implies the existence of a non-Lebesgue-measurable set of real numbers of type $ \Sigma_{1}^{2} $ (see [6]), the negation of the Suslin Hypothesis and the non-existence of measurable cardinals (see [2]).

References

[1a] K. Gödel, “The consistency of the axiom of choice and of the generalized continuum hypothesis”, Proc. Nat. Acad. Sci. USA, 24 (1938), pp. 556–557.
[1b] K. Gödel, “Consistency proof for the generalized continuum 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 $ \mathsf{ZF} $-set theory”, Springer (1971).

Comments

Concerning (the notation) $ \Sigma_{1}^{2} $, see Descriptive Set Theory.

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

New results can be found in [a1], which is also a good introduction to the concept of 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.
How to Cite This Entry:
Gödel constructive set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%B6del_constructive_set&oldid=23312
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article