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Feb 13, 2000, 3:00:00 AM2/13/00

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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.

--

2

z = z + c

n + 1 n

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.

--

2

z = z + c

n + 1 n

c:\dos

c:\dos\run

run\dos\run

Feb 24, 2000, 3:00:00 AM2/24/00

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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.

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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