Countable vs Uncountable Infinity
The cluster focuses on debates about countable and uncountable infinite sets, Cantor's diagonal argument proving the reals are uncountable, and cardinality comparisons between natural numbers, rationals, and reals.
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Not just infinite, uncountably infinite!
It is a precise term with a precise meaning. You can look it up and found many, many explanations online, some of which will be right, few of which will be truly helpful.Let me add to them.If you can put a set into one-to-one correspondence with the counting number (which are 1, 2, 3, and so on) then we call the set "countable" or "countably infinite". Examples include (but are not limited to):* The even counting numbers: 2, 4, 6, 8, 10, and so on;* The integer
Oops, that's what I meant, not even countably in-finite
I thought Aleph Null was countably infinite?
Cantor's diagonal argument -- proof that the Real numbers are more infinite than the Integers (or Naturals). Slightly more mind-blowing than the algebraics are no bigger than the naturals.
You seem to misunderstand the concept of countable infinity
It's not the set of digits you use which is uncountable, it's the representation itself.
Countably infinite? :)https://en.wikipedia.org/wiki/Countable_set
Most of them. The reals are uncountable. The set of finite expressions is countable.
Not all infinities are equal, from https://en.wikipedia.org/wiki/Cardinal_number:"A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers."