[FOM] Is CH vague?
pratt at cs.stanford.edu
Sun Feb 3 15:46:43 EST 2008
laureano luna wrote:
> I'd say the arrow of entailment goes from axioms to
> theorems, as logicians demand, though the arrow of
> causality (motivation) goes often the other way
I have two comments on this.
First, in many contexts it is the theory that is of primary interest,
the existence of one or another axiomatization is only secondary.
Optimizing the axiomatization to suit changing needs moves the arrow of
entailment around, revealing its fickle nature.
Second, where does one draw the line between axioms and hypotheses?
Proving P --> Q from a set G of axioms is the same thing (with the
appropriate caveats) as proving Q from the set G,P of axioms. When P
arises frequently in the role of hypothesis it is convenient to promote
it to an axiom. In that context P can be said to be axiomatic, and
after it has held that status sufficiently long the possibility of not-P
starts to sound like madness, making P a religion worth fighting for.
In extreme cases no holds are barred, witness 9/11, fortunately it has
not come to that for set theory vs. category theory.
Those able to remember or reconstruct the more humble origin of the
axiom as an hypothesis can see that the interpretation of "axiomatic" as
"necessary" is illusory.
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