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No elementary embedding from v into v is definable from parameters

Published online by Cambridge University Press:  12 March 2014

Akira Suzuki*
Affiliation:
Department of Mathematics, School Of Science, Nagoya University, Chikusa-KU, Nagoya, 464-8602, Japan, E-mail: akira@math.nagoya-u.ac.jp

Extract

In 1970, Kenneth Kunen showed that there is no non-trivial elementary embedding of the universe V into itself [2] using the axiom of choice. Kunen remarked in his paper that the result can be formalized in Morse-Kelley theory of sets and classes. In this paper, we will work within ZF, Zermelo-Fraenkel axioms, and deal with embeddings definable with a formula and a parameter.

In ZF, a “class” is usually synonymous with “property”, that is a class definable with a parameter, C = {x: φ(x,p)}, where φ is a formula in the language [∈}. Using this convention, let j be a class. Then “j is an elementary embedding of V into V” is not a single statement but a schema of statements “j preserves ψ” for each formula ψ. We prove that this schema is expressible in the language {∈} by a single formula:

Lemma. An embedding j: V → V is elementary iff j preservesψ.

Here ψ(α, ψ, a) is the property “a is an ordinal, φ is a formula and Vα.”

The lemma is of course a schema of lemmas, one for each formula denning j and for each ψ to be preserved.

Using this we prove our theorem in ZF (again, a schema of theorems.):

Theorem 1.1. There is no nontrivial definable elementary embedding j: V → V.

Many symbols and their definitions follow those used by Drake's book [1]. The formula Sat expresses the satisfaction relation . The formula Fmla(u) expresses that u is the Gödel-set for a formula.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1]Drake, F. R., Set Theory: An Introduction to Large Cardinals, North-Holland, Amsterdam, London, 1974.Google Scholar
[2]Kunen, K., Elementary embeddings and infinitary combinatorics, this Journal, vol. 36 (1971).Google Scholar
[3]Vickers, J. and Welch, P. D., On Elementary Embeddings from an Inner Model to the Universe, to appear.Google Scholar
[4]Woodin, H., absoluteness and supercompact cardinals, 05 15 1985, hand-written note.Google Scholar