Английская Википедия:Gödel operation

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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. Шаблон:Harvs 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

Шаблон:Harvtxt used the following eight operations as a set of Gödel operations (which he called fundamental operations):

1. <math>\mathfrak{F}_1(X,Y) = \{X,Y\}</math>
2. <math>\mathfrak{F}_2(X,Y) = E\cdot X = \{(a,b)\isin X\mid a\isin b\}</math>
3. <math>\mathfrak{F}_3(X,Y) = X-Y</math>
4. <math>\mathfrak{F}_4(X,Y) = X\upharpoonright Y= X\cdot (V\times Y) = \{(a,b)\isin X\mid b\isin Y\}</math>
5. <math>\mathfrak{F}_5(X,Y) = X\cdot \mathfrak{D}(Y) = \{b\isin X\mid\exists a (a,b)\isin Y\}</math>
6. <math>\mathfrak{F}_6(X,Y) = X\cdot Y^{-1}= \{(a,b)\isin X\mid(b,a)\isin Y\}</math>
7. <math>\mathfrak{F}_7(X,Y) = X\cdot \mathfrak{Cnv}_2(Y) = \{(a,b,c)\isin X\mid(a,c,b)\isin Y\}</math>
8. <math>\mathfrak{F}_8(X,Y) = X\cdot \mathfrak{Cnv}_3(Y)= \{(a,b,c)\isin X\mid(c,a,b)\isin Y\}</math>

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, and so on.

Шаблон:Harvtxt uses the following set of 10 Gödel operations.

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

Properties

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

References

• Шаблон:Cite book
• Шаблон:Citation