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 ''G''₁, ''G''₂,...

Definition

used the following eight operations as a set of Gödel operations (which he called fundamental operations):
#$\mathfrak_1(X,Y) = \$
#$\mathfrak_2(X,Y) = E\cdot X = \$
#$\mathfrak_3(X,Y) = X-Y$
#$\mathfrak_4(X,Y) = X\upharpoonright Y= X\cdot (V\times Y) = \$
#$\mathfrak_5(X,Y) = X\cdot \mathfrak(Y) = \$
#$\mathfrak_6(X,Y) = X\cdot Y^= \$
#$\mathfrak_7(X,Y) = X\cdot \mathfrak_2(Y) = \$
#$\mathfrak_8(X,Y) = X\cdot \mathfrak_3(Y)= \$
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.

uses the following set of 10 Gödel operations.

#$G_1(X,Y) = \$
#$G_2(X,Y) = X\times Y$
#$G_3(X,Y) = \$
#$G_4(X,Y) = X-Y$
#$G_5(X,Y) = X\cap Y$
#$G_6(X) = \cup X$
#$G_7(X) = \text(X)$
#$G_8(X) = \$
#$G_9(X) = \$
#$G_(X) = \$

Properties

Gödel's normal form theorem states that if φ(''x''₁,...''x''_''n'') is a formula in the language of set theory with all quantifiers bounded, then the function {(''x''₁,...,''x''_''n'') ∈ ''X''₁×...×''X''_''n'' , φ(''x''₁, ..., ''x''_''n'')) of ''X''₁, ..., ''X''_''n'' is given by a composition of some Gödel operations. This result is closely related to Jensen's rudimentary functions.K. Devlin
An introduction to the fine structure of the constructible hierarchy
(1974, p.11). Accessed 2022-02-26.

References

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Inline references

{{DEFAULTSORT:Godel operation
Constructible universe