rationals - 1 EXTREMELY LARGE CARDINALS IN THE RATIONALS by...

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1 EXTREMELY LARGE CARDINALS IN THE RATIONALS by Harvey M. Friedman Department of Mathematics Ohio State University August, 1996 friedman@math.ohio-state.edu In 1995 we gave a new simple principle of combinatorial set theory and showed that it implies the existence of a nontrivial elementary embedding from a rank into itself, and follows from the existence of a nontrivial elementary embedding from V into M, where M contains the rank at the first fixed point above the critical point. We then gave a “diamondization” of this principle, and proved its relative consistency by means of a standard forcing argument. We have recently discovered how to pull this diamondization down into the rationals in a natural and simple way using the concept of first order definability. This results in a Π -0-1 sentence which implies the consistency of ZFC + the existence of a nontrivial elementary embedding of a rank into a rank, and which follows from the consistency of ZFC + the existence of a nontrivial elementary embedding from V into M, where M contains the rank at the first fixed point above the critical point. Here are the details. First we state the extremely large cardinal hypotheses commonly called I1, I2, and I3: I1. There is a nontrivial elementary embedding from some V( α +1) into V( α +1). I2. There is a nontrivial elementary embedding from V into a transitive class M such that V( λ ) M, where λ is the first fixed point after the critical point. I3. There is a nontrivial elementary embedding from some V( α ) into V( α ). See The Higher Infinite, Aki Kanamori, Perspectives in Mathematical Logic, Springer-Verlag, 1994, p. 325 for a discussion, where it is shown that I1 implies I2 implies I3, and in fact I1 implies the consistency of I2, and I2 implies the consistency of I3.
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rationals - 1 EXTREMELY LARGE CARDINALS IN THE RATIONALS by...

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