Gödel operation

In mathematical set theory, a set of Gödel operations is a finite collection of operations on sets that can be used to construct the constructible sets from ordinals. introduced the original set of 8 Gödel operations 𝔉1,...,𝔉8 under the name fundamental operations. Other authors sometimes use a slightly different set of about 8 to 10 operations, usually denoted G1, G2,...

Definition
used the following eight operations as a set of Gödel operations (which he called fundamental operations): The second expression in each line gives Gödel's definition in his original notation, where the dot means intersection, V is the universe, E is the membership relation, $$\mathfrak{D}$$ denotes range and so on. (Here the symbol $$\upharpoonright$$ is used to restrict range, unlike the contemporary meaning of restriction.)
 * 1) $$\mathfrak{F}_1(X,Y) = \{X,Y\}$$
 * 2) $$\mathfrak{F}_2(X,Y) = E\cdot X = \{(a,b)\isin X\mid a\isin b\}$$
 * 3) $$\mathfrak{F}_3(X,Y) = X-Y$$
 * 4) $$\mathfrak{F}_4(X,Y) = X\upharpoonright Y= X\cdot (V\times Y) = \{(a,b)\isin X\mid b\isin Y\}$$
 * 5) $$\mathfrak{F}_5(X,Y) = X\cdot \mathfrak{D}(Y) = \{b\isin X\mid\exists a (a,b)\isin Y\}$$
 * 6) $$\mathfrak{F}_6(X,Y) = X\cdot Y^{-1}= \{(a,b)\isin X\mid(b,a)\isin Y\}$$
 * 7) $$\mathfrak{F}_7(X,Y) = X\cdot \mathfrak{Cnv}_2(Y) = \{(a,b,c)\isin X\mid(a,c,b)\isin Y\}$$
 * 8) $$\mathfrak{F}_8(X,Y) = X\cdot \mathfrak{Cnv}_3(Y)= \{(a,b,c)\isin X\mid(c,a,b)\isin Y\}$$

uses the following set of 10 Gödel operations.


 * 1) $$G_1(X,Y) = \{X,Y\}$$
 * 2) $$G_2(X,Y) = X\times Y$$
 * 3) $$G_3(X,Y) = \{(x,y)\mid x\isin X, y\isin Y, x\isin y\}$$
 * 4) $$G_4(X,Y) = X-Y$$
 * 5) $$G_5(X,Y) = X\cap Y$$
 * 6) $$G_6(X) = \cup X$$
 * 7) $$G_7(X) = \text{dom}(X)$$
 * 8) $$G_8(X) = \{(x,y)\mid(y,x)\isin X\}$$
 * 9) $$G_9(X) = \{(x,y,z)\mid(x,z,y)\isin X\}$$
 * 10) $$G_{10}(X) = \{(x,y,z)\mid(y,z,x)\isin X\}$$

Properties
Gödel's normal form theorem states that if φ(x1,...xn) is a formula in the language of set theory with all quantifiers bounded, then the function {(x1,...,xn) ∈ X1×...×Xn | φ(x1, ..., xn)) of X1, ..., Xn is given by a composition of some Gödel operations. This result is closely related to Jensen's rudimentary functions.