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element of S after a’.
The statement M[S] = j(x,>,>> x)) can be viewed as a
statement in V(»S) about x,b. Likewise, the statement M[S]
= j(x,>,>> y) can be viewed as a statement in V(»S) about
x,b’. By property ii), the two statements must have the same
truth value. QED
THEOREM 7.7. Basic + Minimal + Existence + Amplification
has the same Tarski degree as ZF. I.e., they are mutually
interpretable. 10
For Basic + Minimal + Existence + Binary Amplification, we
need S of type w satisfying a more powerful form of
indiscernibility.
LEMMA 7.8. Suppose there is a countable transitive model of
ZC + “there exists a measurable cardinal”. There is a
countable transitive model A of ZF, and an unbounded S Õ A
consisting of limit ordinals, of order type w, with the
following indiscernibility property. Let a < b < g be from
S. Then S\b and S\g have the same first order properties
over A, relative to any parameters from the V(a) of A.
Proof: Inside the given model, shoot a Prikry sequence of
limit ordinals through the measurable cardinal, l. Take A to
be the V(l) of the given model, and S to be the range of the
Prikry sequence. QED
Let A be a transitive model of ZF, and S be an unbounded
subset of A consisting of limit ordinals. Define M[A,S] as
follows. The domain and the > is defined internally in M,
as proper classes of M, as before. We write these as DM, >M.
The >>, which is a binary relation on DM, is then defined as
before. We write this as >>S.
LEMMA 7.9. M[A,S] satisfies Basic + Minimality + Existence.
LEMMA 7.10. Let A,S be as in the conclusion of Lemma 7.8.
Then M[A,S] satisfies Binary Amplification.
Proof: Let
the binary
element of
of S after
Œ DMa’. Let y >M x, and j(x) true in M[A,S], using >M,=, and
relation y >>S z >>S x. Let a be the least
S such that x Œ DMa. Let b be the least element
a. Let a’ be the least element of S such that y
b’ be the least element of S after a’. Let j(x) be a statement in M[A,S] about x Œ DM, using >M,
and the binary relation z >>S w >>S x. The statement M[A,S]
= j(x,>,binary >> x,=)) can be viewed as a statement in
(M,Œ,S\b) about x. This is because the binary relation z >>S
w >>S x can be defined from M and S\b. Likewise, the
statement M[A,S] = j(x,>,binary >> y,=) can be viewed as
the corresponding statement in (M,Œ,S\b’) about x. By the
indiscernibility property, the two statements must have the
same truth value. QED
THEOREM 7.11. Basic + Minimality + Existence + Binary
Amplification interprets ZFC + “there exists a Ramsey 11
cardinal” and is interpretable in ZFC + “there exists a
measurable cardinal”.
PRINCIPLE OF PLENITUDE
From Wikipedia, Plenitude Principle.
The principle of plenitude asserts that everything that can
happen will happen.
The historian of ideas Arthur Lovejoy was the first to
discuss this philosophically important Principle
explicitly, tracing it back to Aristotle, who said that no
possibilities which remain eternally possible will go
unrealized, then forward to Kant, via the following
sequence of adherents:
Augustine of Hippo brought the Principle from NeoPlatonic
thought into early Christian Theology.
St Anselm 's ontological arguments for God's existence used
the Principle's implication that nature will become as
complete as it possibly can be, to argue that existence is
a 'perfection' in the sense of a completeness or fullness.
Thomas Aquinas's belief in God's plenitude conflicted with
his belief that God had the power not to create everything
that could be created. He chose to constrain and ultimately
reject the Principle.
Giordano Bruno's insistence on an infinity of worlds was
not based on the theories of Copernicus, or on observation,
but on the Principle applied to God. His death may then be
attributed to his conviction of its truth.
Leibniz believed that the best of all possible worlds would
actualize every genuine possibility, and argued in
Théodicée that this best of all possible worlds will
contain all possibilities, with our finite experience of
eternity giving no reason to dispute nature's perfection.
Kant believed in the Principle but not in its empirical
verification, even in principle.
The Infinite monkey theorem and Kolmogorov's zeroone law
of contemporary mathematics echo the Principle. It can also
be seen as receiving belated support from certain radical 12
directions in contemporary physics, specifically the manyworlds interpretation of quantum mechanics and the
cornucopian speculations of Frank Tipler on the ultimate
fate of the universe.
REFERENCE
[1] H. Friedman, Concept Calculus, Preprints, #53,
http://www.math.ohiostate.edu/%7Efriedman/manuscripts.html
*This research was partially supported by NSF Grant DMS
0245349....
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 Fall '08
 JOSHUA
 Math, Calculus, Set Theory

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