[FOM] 602: Removing Deep Pathology 2

[martdowd at aol.com]

 Harvey Friedman writes:



THEOREM. There is no formula in the language
of set theory such that
ZFC proves defines a discontinuous solution to f(x+y) =
f(x) + f(y).
Furthermore, this is true even with parameters for real numbers in


 

 The terminology "deeply pathological" has a subjective connotation.

http://math.stackexchange.com/questions/166176/a-question-concerning-on-the-axiom-of-choice-and-cauchy-functional-equation/
states some results on non-continuos solutions to the Cauchy functional
equation.  They need not exists, but do if full choice holds.

The theorem stated by Harvey "classifies" these counterexamples as
non-definable, a least unless V=L is assumed.

see also
http://mathoverflow.net/questions/46063/explicit-hamel-basis-of-real-numbers

- Martin Dowd


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