David Welch welch at
Tue Apr 3 18:34:44 MDT 2001

On Mon, Apr 02, 2001 at 11:37:48AM -0400, Charles Brown wrote:
> What about Russell's paradox and Goedel's proof , both of the 20th Century ?
Russell's paradox and Godel's incompleteness theorems don't (obviously)
concern the relation between mathematics and reality, rather the relationship
between mathematical practice and its logical/philosophical formalisation.
The former demonstrates that set theory using the Axiom of Abstraction is
inconsistent but all later axiomatisations of set theory have preserved
the Axiom of Abstraction in essence while avoiding the paradoxes. In
particular ZFC is so widely accepted that I don't believe any mathematician
worries about her use of set theory. Similarly, Godel's incompleteness
theorems demonstrate the impossibility of completing Hilbert's conservation
and consistency programs but since Hilbert's programs are only of historical
interest anyway the impact on what mathematicians do is very limited.

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