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WORKING WITH NONSTANDARD MODELS
Harvey M. Friedman
[email protected]
http://www.math.ohiostate.edu/~friedman/
July 31, 2003
Most of the research in foundations of mathematics that I
do in some way or another involves the use of nonstandard
models. I will give a few examples, and indicate what is
involved.
1. General algebra and measurable cardinals. An
unexpectedly direct connection.
2. Borel selection and higher set theory. A descriptive set
theoretic context extensively pursued by some functional
analysts.
3. Equational Boolean relation theory and Mahlo cardinals.
A discrete mathematical context.
4. We conclude with an adaptation of 1 to weak second order
logic.
1. GENERAL ALGEBRA AND MEASURABLE CARDINALS.
Some innocent looking statements in general algebra turn
out to be equivalent to the existence of a measurable
cardinal. In fact, the first measurable cardinal turns out
to be clearly identifiable in a basic context in general
algebra.
Here an algebra is just a relational structure based on
finitely many constant and function symbols and no relation
symbols.
PROPOSITION 1.1. Every algebra with a sufficiently large
domain has a proper extension with the same countable
subalgebras up to isomorphism.
PROPOSITION 1.2. Every algebra with a sufficiently large
domain has a proper extension with the same finitely
generated subalgebras up to isomorphism.
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THEOREM 1.3. Propositions 1.1 and 1.2 are provably
equivalent to the existence of a measurable cardinal, over
ZFC.
THEOREM 1.4. The least cardinal
k
, if any, such that every
algebra of cardinality
≥
k
has a proper extension with the
same countable (alternatively finitely generated)
subalgebras up to isomorphism, is the least measurable
cardinal.
Note that these Propositions and Theorems use the notion of
cardinality, which can be argued to be not strictly
algebraic. We give obvious reformulations which do not use
the notion of cardinality.
PROPOSITION 1.1’. Every algebra with a sufficiently
inclusive domain has a proper extension with the same
countable subalgebras up to isomorphism.
PROPOSITION 1.2’. Every algebra with a sufficiently
inclusive domain has a proper extension with the same
finitely
generated subalgebras up to isomorphism.
THEOREM 1.3’. Propositions 1.1’ and 1.2’ are provably
equivalent to the existence of a measurable cardinal, over
ZFC.
THEOREM 1.4’. Let D be a nonempty set. The following are
equivalent.
i) every algebra whose domain includes D has a proper
extension with the same countable subalgebras up to
isomorphism;
ii) every algebra whose domain includes D has a proper
extension with the same finitely generated subalgebras up
to isomorphism;
iii) there is a countably additive 0,1 valued measure on
all subsets of D, in which singletons have measure 0 and D
has measure 1.
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 Fall '08
 JOSHUA
 Math, Model theory, Lemma, Borel, measurable cardinal, proper extension, borel selection

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