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SOME HISTORICAL PERSPECTIVES ON CERTAIN INCOMPLETENESS
PHENOMENA
Harvey M. Friedman
Department of Mathematics
Ohio State University
[email protected]
www.math.ohiostate.edu/~friedman/
May 21, 1997
We have been particularly interested in the demonstrable
unremovability of machinery, which is a theme that can be
pursued systematically starting at the most elementary level
 the use of binary notation to represent integers; the use
of rational numbers to solve linear equations; the use of
real and complex numbers to solve polynomial equations; and
the use of transcendental functions to solve differential
equations.
Practical situations arise such as the use of complex
variables in number theory, or group theory in topology.
Here there has been no demonstrable
unremovability.
But it appears that when the machinery is removed, clarity
and power is lost. This kind of unremovability is
extremely
difficult to get at rigorously.
Over the years, a growing collection of cases of
demonstrable unremovability of increasing interest have been
developed. Here is a brief synopsis of some of the
highlights.
Around 1967, Tony Martin solved a crucial problem in
infinite game theory involving Borel sets, using a massive
amount of machinery, going well beyond the usual axioms for
mathematics.
Around 1968, we proved that a small part of the machinery
was unremovable 
uncountably many
uncountable cardinals.
In 1974, after an extended effort, Martin reduced the amount
of machinery he used to
uncountably many
uncountable cardinals.
We later gave the following reinterpretation of Martin's
theorem:
every Borel set in the plane that is symmetric
about the line y = x contains or is disjoint
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from the graph of a Borel function,
and showed the unremovability of
uncountably many uncountable cardinals
for this statement.
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 Fall '08
 JOSHUA
 Math, Set Theory, Topology, Borel, infinite sequence

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