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:::2. The theorem states that if R meets these conditions, then R is not countable without defining countable. If one accepts that one can first define '''not countable''' and then determine '''countable''' as its '''compliment''', then it might appear that Cantor's first ideas had nothing to do with the notion of a '''one-to-one correspondence of a subset with its proper subset''' as the definition of countable.
:::3. The theorem could be stated as follows: '''Any set that has at least these 4 properties is not countable'''. But to state it this way, is the same as saying the real numbers are not countable by virtue of their '''design/architecture'''.
:::4. In set theory, if a proper subset contains a certain property, then it implies the superset must also posses that property by '''inclusion''' of the subset. So if the rational numbers are countable, this would imply the real numbers must therefore also be countable.
Or is it that you simply can't bear to admit Cantor's theorem and his theories in general are cranky?
Hardy, I really don't want you becoming upset over this. Just take a deep breath and stay calm. Try to be your normal, cool-headed, objective self.
[[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 09:14, 22 January 2010 (UTC)
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