Zermelo-Fraenkel Set Theory
المؤلف:
Montague, R.
المصدر:
"Semantic Closure and Non-Finite Axiomatizability. I." In Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, (Warsaw, 2-9 September 1959). Oxford, England: Pergamon
الجزء والصفحة:
...
30-12-2021
1917
Zermelo-Fraenkel Set Theory
A version of set theory which is a formal system expressed in first-order predicate logic. Zermelo-Fraenkel set theory is based on the Zermelo-Fraenkel axioms.
Zermelo-Fraenkel set theory is not finitely axiomatized. For example, the axiom of replacement is not really a single axiom, but an infinite family of axioms, since it is preceded by the stipulation that it is true "For any set-theoretic formula 
." Montague (1961) proved that Zermelo-Fraenkel set theory is not finitely axiomatizable, i.e., there is no finite set of axioms which is logically equivalent to the infinite set of Zermelo-Fraenkel axioms. von Neumann-Bernays-Gödel set theory provides an equivalent finitely axiomized system.
REFERENCES:
Montague, R. "Semantic Closure and Non-Finite Axiomatizability. I." In Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, (Warsaw, 2-9 September 1959). Oxford, England: Pergamon, pp. 45-69, 1961.
Zermelo, E. "Über Grenzzahlen und Mengenbereiche." Fund. Math. 16, 29-47, 1930.
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