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Computable Numbers

Name: Anonymous 2010-04-10 22:01

Almost all real numbers are non-computable.
Blows my mind.
Discuss.

Name: Anonymous 2010-04-17 19:49

>>2

Yes, almost all.

On a basic level, there are many different infinite cardinalities, so for example any countably infinite set is neglible in an uncountable one, and so it's complement can be considered to be 'almost all'

In a more precise way measure theory allows us to define sets of measure 0 and so a statement P(x) is said to hold 'almost always' if {x : not P(x)} has measure 0.

So for example, even an uncountable set can be neglible in an uncountable set with the 'obvious; (Lesbesgue) measure, consider for example the cantor set.

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