An Elementary Math Lesson for All People Like "Democratic Grand Demon" and [Jiucai]!

A problem that anyone who attended elementary school math would know about -- ordinarily it wouldn't warrant a dedicated main post. But I've discovered that this problem isn't limited to just one person; it's actually quite widespread. People don't even realize they're wrong and are all quite self-righteous about it. So it seems this elementary math lesson must be given to such people. Let me first quote two exemplary errors:

In English, "uncountable" and "innumerable" do have the same meaning. From the perspective of empiricist philosophy, uncountable means innumerable. Do you understand philosophy? (Democratic Grand Demon: 2006-08-18 10:09:57)0B (0/13/1)
[Jiucai] posted on 2006-08-18 13:19:12
You can't prove that "human thoughts" and "human ideas" are countable, so just honestly admit it! Otherwise you lack academic spirit! That's what your Miss Chan said. The interval between each person's thoughts and ideas cannot be infinitely short, correct! But human thoughts and ideas cannot be quantified either, that's also a fact, right? How do you count something that can't be quantified?

As you can see, both of the above self-righteously equate "countable" and "uncountable" with "can be enumerated" and "cannot be enumerated," and they still bring up philosophy and academics, as if they're not the least bit afraid of being laughed at. Truly, one grain of rice feeds a hundred kinds of people -- no surprise there.

But in science, countable and uncountable have the strictest mathematical definitions, something known even in elementary school math:
Countable: can be put in one-to-one correspondence with the natural numbers or a subset thereof.
Uncountable: cannot be put in one-to-one correspondence with the natural numbers or a subset thereof.

Countable versus uncountable has absolutely nothing to do with everyday "can be enumerated" or "cannot be enumerated." Just as no one can enumerate all rational numbers, yet the rational numbers are countable; just as at a time when no one could find a single transcendental number, we could already clearly prove that transcendental numbers are uncountable. Scientific concepts are not marketplace thinking. People who don't even know the most basic scientific concepts still talk about philosophy and academics -- is it that certain parts of their anatomy are excessively thick?

Finite necessarily implies countable -- this is also a concept known from elementary school math. Why? Because finite necessarily corresponds one-to-one with some subset of the natural numbers. But countable does not necessarily mean finite, just as the rational numbers mentioned above are countable but infinite. These issues have nothing to do with Miss Chan's proof; they are all the most basic scientific concepts. Please stop making a laughingstock of yourselves!