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AcademicGround >> Proof -Cantor Continuum Hypothesis


8/6/04 1:41 PM
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Rastus
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Edited: 09-Aug-04 05:04 PM
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I was corresponding with jgibson recently. The problem we were working on reminded me of a "proof" for the Continuum Hypothesis I thunk up a couple of years ago. For those hardcore math guys (or interested parties), here is a sketch: Cantor's Continuum Hypothesis: Sketch of Proof The cardinality of the real #'s is the next higher infinity to aleph zero, the cardinality of the natural numbers. Given: 2^Aleph0 is the next higher cardinality to Aleph0. In other words, the set of all ORDERED natural numbers is the set with next highest cardinality. The proof is to create one-to-one mappings between a subset of the reals, [0.1 ... 1.0...], and the set of ordered natural numbers. This solution assumes the Axiom of Choice. let a=sequence of integers = a1, a2, ... Since we know the set of reals is greater than aleph0, it is trivial to note that |2^aleph0| <= |Reals|. The trick is showing a one-one map between the set of reals and the set of ordered natural numbers. This mapping would prove they’re equal. In other words, for any r in our segment between 0.1 and 1.0, there exists one and only one sequence of integers, a, such that f( r ) = a. Look at irrational numbers in (0.1 ... 1.0) (line segment between 0.1 and 1) as follows: Any irrational number will have an infinite number of decimal places. Suppose we construct a sequence of integers off of those decimal places according to the following rules: Concatenate numbers until the number obtained is greater than the previous number. Our number cannot end if the next digit is a zero. e.g.'s 1. r = 0.191111291.... a1 = 1
a2 = 9
a3 = 11
a4 = 112
etc... so we could CHOOSE a function, x, such that x(r)=a, where a = (1,9,11,112,...) and r = 0.191111291... You can see how the axiom of choice is critical in this proof. example 2. r=0.10120005... a1 = 10
a2 = 12000
a3 = ... In other words, given any r in R, there exists a function x (selective concatenation) that takes a set of irrational numbers into R on a one-to-one basis. That means that the cardinality of R is less than or equal to that of the ordered natural numbers, thereby proving Cantor's Continuum Hypothesis.
8/9/04 1:12 PM
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Crazy Zimmerman
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Edited: 09-Aug-04
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All this "proves" is that you are a math nerd.
8/9/04 5:18 PM
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Rastus
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Edited: 09-Aug-04
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Lord, I'm getting heckled on the AcademicGround? Maybe I should have posted this on the OG... *thinks for half a second* nah.... Don't we have any math guys out there who are familiar with this set theory? I know Dogbert is.
8/10/04 10:02 AM
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Crazy Zimmerman
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Edited: 10-Aug-04
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LOL. Sorry, I couldn't resist. I found this ground by mistake obviously, and highly doubt I will ever return. --Not So Academic Zimmerman
8/10/04 1:45 PM
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Rastus
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Edited: 10-Aug-04
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Don't hate...masticate. - Over Eater's Club
8/10/04 9:25 PM
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Dogbert
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Edited: 10-Aug-04
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I will look at it again.
8/11/04 8:41 PM
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Mhorad
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Edited: 11-Aug-04
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interesting

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