The refutation of axioms

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Feb 13, 2000, 3:00:00 AM2/13/00
Every good mathematician knows that any formal proof is based on axioms.
These axioms can not be reduced any further. They are just the "obvious
truths", like ~Ex.(x>x) or Ax.[Ea.(x=a)]. I'm thinking about how the people
who founded non-Euclidean geometry got to it by refuting one of the axioms,
to be more specific, the parral postulate (please pardon my spelling). You
could take any system and think up another system by refuting one or more
of it's axioms. For example, suppose you said, "What if there did exist a
number x in which x>x?". You could make a new system based on that. And
just like non-Euclidean geometry, it may have a basis in reality. Maybe
inside black holes x>x. I don't know.
z = z + c
n + 1 n


Richard A. Beldin

Feb 24, 2000, 3:00:00 AM2/24/00
Axioms are not "obvious truths" but "conventional" starting points for
argument. We gave up the notion of "obvious truths" many years ago. To get a
new axiom for a system you would need to choose a statement which is
independent of the current system if you wish to include the theorems you
already have proven. If you want a system which excludes some of those
theorems, then you can choose an alternative conclusion to a hypothesis.
However, anything as drastic as "x > x" destroys the conventional meaning of
">". To work in such a system, you would have to unlearn too much for that to
be a useful axiom.
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