Content deleted Content added
Line 270:
:
That is nonsense. The set of all real algrebraic numbers contains the set of all rational numbers, but does not satisfy property 4 at all. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 05:51, 22 January 2010 (UTC)
::It's quite amazing to me how you picked out what you felt like answering and ignored the big elephant in the room. So, I respond as follows: I posted a second comment that clarified my first assumption. Your example of a finite and infinite set is irrelevant - all the sets we are discussing are infinite: naturals, rationals and reals. So how about addressing the real issues: :::1. Stating the 4 properties, says that R contains real numbers.
:::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. [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 09:14, 22 January 2010 (UTC)
| |||