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Confused By What You Mean By Completeness

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Posted by Richard Ruquist on May 27, 2003 15:48:23 UTC


Paul,

I was with you until you said that 5 divided by 4 is not in the set of integers and the set is therefore not complete.

What Goedel seemed to be saying was just the opposite, that true relationships exist in incomplete systems of numbers that cannot be derived from the axioms of the system.

It's like saying that 5/4 can be found in the set of integers, but cannot be derived in that set.

From what you said, every system of numbers is incomplete. For example, the set of all real numbers would be incomplete as it does not include irrational numbers or even imaginary numbers.

Are there true relationships in the set of integers that cannot be derived within that set?

yanniru

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