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Natural internal forcing schemata extending ZFC: Truth in the universe?

Published online by Cambridge University Press:  12 March 2014

Garvin Melles
Garvin Melles*
Affiliation:
Department of Mathematics, Bar Ilan University, Ramat Gan, Israel Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
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Extract

Mathematicians have one over on the physicists in that they already have a unified theory of mathematics, namely, set theory. Unfortunately, the plethora of independence results since the invention of forcing has taken away some of the luster of set theory in the eyes of many mathematicians. Will man's knowledge of mathematical truth be forever limited to those theorems derivable from the standard axioms of set theory, ZFC? This author does not think so, he feels that set theorists' intuition about the universe of sets is stronger than ZFC. Here in this paper, using part of this intuition, we introduce some axiom schemata which we feel are very natural candidates for being considered as part of the axioms of set theory. These schemata assert the existence of many generics over simple inner models. The main purpose of this article is to present arguments for why the assertion of the existence of such generics belongs to the axioms of set theory.

Our central guiding principle in justifying the axioms is what Maddy called the rule of thumb maximize in her survey article on the axioms of set theory, [8] and [9]. More specifically, our intuition conforms with that expressed by Mathias in his article What is Maclane Missing? challenging Mac Lane's view of set theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 2 , June 1994 , pp. 461 - 472
Copyright
Copyright © Association for Symbolic Logic 1994

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References

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