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Judith,
I am not sure if
set theory, including infinite sets, is nonformalizable. I believe it is
nonformalizable. If so, it contain's irremovable semantic aspects. (The diagonal
argument itself is one that strikes me as being of a transcendental (in the
mathematical sense) nature.)
Regards,
Tim
Hi Tim,
In each of his books, my father wrote at length about what
happens in mathematics when all semantics are removed and replaced with purely
syntactical rules. His belief was that mathematics is a complex system in its
own right. That means mathematics as a system is relational and has
its own organization which we cannot understand when we fractionate it or
reduce it. That means that the inherent semantics generated by such
organization is indispensable. When Hillbert, et al, tried to "improve
mathematics" by divorcing the system from all semantics, it deformed
mathematics into a system characterized by a weird collection
of paradoxes which make it useless as a modeling tool for much of anything.
Judith
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