[hist-analytic] The status and relevance of "standard logic"
danny.frederick at btinternet.com
Sat Oct 31 12:44:05 EDT 2009
The metamathematical proofs to which you refer have hidden axioms that are
far from obvious. In order to avoid paradoxes which refute the whole system,
they typically have a theory of logical or linguistic types which is built
into the formation rules for formulae of the system. Such type-theories are
a lot more doubtful than the truths of elementary arithmetic.
Besides, the question of how doubtful something appears does not bear upon
its truth. Frege was one of the greatest mathematicians. His Basic Law V
seemed self-evident to him. Yet it was shown by Russell to be logically
false. As Russell stated in PM (second edition, p.59) we have no guarantee
that another paradox will not surface however happy we may be with the
latest set of axioms (open and hidden).
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