Se p 19 92 Natural Internal Forcing Schemata extending ZFC . A Crack in the Armor surrounding CH ?

Mathematicians are 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, ZF C? This author does not think so, and in fact he feels there is a schema concerning non-constructible sets which is a very natural candidate for being considered as part of the axioms of set theory. To understand the motivation why, let us take a very short look back at the history of the development of mathematics. Mathematics began with the study of mathematical objects very physical and concrete in nature and has progressed to the study of things completely imaginary and abstract. Most mathematicians now accept these objects as as mathematically legitimate as any of their more concrete counterparts. It is enough that these objects are consistently imaginable, i.e., exist in the world of set theory. Applying the same intuition to set theory itself, we should accept as sets as many that we can whose existence are consistent with ZF C. Of course this is only a vague notion, but knowledge of set theory so far, namely of the existence of L provides a good starting point. What sets can we consistently imagine beyond L? Since by forcing one can prove the consistency of ZF C with the existence of non-constructible sets and as L is absolute, with these forcing extensions of L you have consistently imagined more sets in a way which satisfies the vague notion mentioned above. The problem is which forcing extensions should * Would like to thank Ehud Hrushovski for supporting him with funds from NSF Grant DMS 8959511