Re: Should we worry about contradictions?


Gc wrote:
|In Peter G. Hinman`s book of logic (2005), it is said that Cantor
based
|his deveploment on to principles: Extensionality and Global
|compherension. Well, we all know that Global compherension leads to
|Russell paradox. I don`t know what is the truth about Cantor`s work
and
|I have read also that opinions like Hinman`s
|are wrong.

By "global comprehension" I assume he means the principle claiming the
existence of a set {x : P(x)} for any property P. I have no idea how
someone could conclude that Cantor believed such a thing. Cantor
described a set as a "collection into a whole" and then did a lot of
straightforwardly mathematical (as opposed to philosophical)
development of the idea, without specifically stated axioms. It was
motivated by his work on Fourier series. He doesn't run into
contradictions. At some point, he realized that sometimes the things
that satisfy P can't be regarded as a "whole" in the same way, e.g.
the ordinals.

Frege built a system having a form of self-contradictory comprehension
in it.

Later, the (controversial) proof of the well-ordering theorem was
analyzed
to find what principles it relied on, and ZFC was developed from that.

Somehow, the simplistic model "naive set theory => contradiction =>
formalized set theory in order to fix the contradiction" seems to have
gotten really deeply ingrained in the literature. I'm actually
somewhat amazed at the persistence of this idea. I think somehow
people find it more exciting to think that set theory has been more
deeply beset by paradoxes than it is or was.

Keith Ramsay

.

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