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The noncommutativity of random and generic extensions

Published online by Cambridge University Press:  12 March 2014

J.K. Truss
J.K. Truss*
Affiliation:
Paisley College of Technology, Paisley, Renfrewshire, Scotland
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Extract

Throughout, M will denote a transitive model of ZFC. Using the terms “random” and “generic” in the sense of [1], one may ask whether there can exist real numbers x and y such that x is generic over M[y] and y is random over M[x]. We shall see below by an elementary argument that this is not possible, and so, in a crude sense at least, random and generic extensions do not commute. This does not however rule out the possibility of a weaker commutativity. Let B be the complete Boolean algebra (in M) for adjoining a random real followed by a generic real and C be the complete Boolean algebra for adjoining a generic real followed by a random real. Then it still might be the case that B and C are isomorphic. This also fails, though, and we shall prove this by establishing the following combinatorial properties of MB and MC:

but

In addition this will show that C cannot be embedded as a complete subalgebra of B.

The property satisfied by B is reminiscent of calibre ℵ1 [2]. B would have calibre if we could replace “infinite” by “uncountable”, and this occurs if Martin's Axiom holds in M. To obtain the nonisomorphism of B and C in general necessitated looking at the weaker property.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 4 , December 1983 , pp. 1008 - 1012
Copyright
Copyright © Association for Symbolic Logic 1983

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References

REFERENCES

[1]Solovay, R.M., A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics (2), vol. 92 (1970), pp. 1–56CrossRefGoogle Scholar
[2]Truss, J.K., Sets having calibre ℵ1, Logic Colloquium 76 (Gandy, R.O. and Hyland, J.M.E., editors), North-Holland, Amsterdam, 1977, pp. 595–612.Google Scholar