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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.ohiostate.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
01 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, SpringerVerlag, 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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 Fall '08
 JOSHUA
 Math

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