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On Σ1 well-orderings of the universe

Published online by Cambridge University Press:  12 March 2014

Leo Harrington  and
Thomas Jech
Leo Harrington
Affiliation:
State University of New York at Buffalo, Amherst, New York 14226
Thomas Jech
Affiliation:
State University of New York at Buffalo, Amherst, New York 14226
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Extract

The constructible universe L of Gödel [2] has a natural well-ordering < L given by the order of construction; a closer look reveals that this well-ordering is definable by a Σ1 formula. Cohen's method of forcing provides several examples of models of ZF + V ≠ L which have a definable well-ordering but none is definable by a relatively simple formula.

Recently, Mansfield [7] has shown that if a set of reals (or hereditarily countable sets) has a Σ1, well-ordering then each of its elements is constructible. A question has thus arisen whether one can find a model of ZF + V ≠ L that has a Σ1 well-ordering of the universe. We answer this question in the affirmative.

The main result of this paper is

Theorem. There is a model of ZF + V ≠ L which has a Σ1 well-ordering.

The model is a generic extension of L by adjoining a branch through a Suslin tree with certain properties. The branch is a nonconstructible subset of ℵ1. Note that by Mansfield's theorem, the model must not have nonconstructible subsets of ω.

Our results can be generalized in several directions. We note that in particular, we can get a model with a Σ1 well-ordering that is not L[X] for any set X. As one might expect from a joint paper by a recursion theorist and a set theorist, the proof consists of a construction and a computation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 1 , March 1976 , pp. 167 - 170
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

Research supported by National Science Foundation grant PO–34191-X02.

References

REFERENCES

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