I've always been hung up on the notion of countable vs uncountable infinity. It seemed a rather silly thing to say, that you could have more than infinity. I understand the argument for it, but from an intuitive level, it felt wrong. Now, I feel that it is completely wrong. There are only two kinds of numbers, those whose digits can be represented and those whose digits can't. 7, 13, 10000000 all belong to the first category as well as .015, .139, and .00000000000000000000000000000000001. PI, 1/3, and e all belong to the second. The second category of numbers are not really numbers but
ideas. They are, in essence, equations. Given a precision, they can produce a number of the first category, but they are not numbers in and of themselves. Another example would be the summation of the reciprocals of all integers, given a precision, we can produce the number, but the summation itself is just an abstract notion. With that we can understand that what we consider to be countable infinity doesn't really exist. It is in the second category and is therefore part of the larger infinity. If you give the countable numbers a precision (limit), then you can discuss those numbers; however, without a limit, the countable numbers are just an idea, much like PI, or the zeta function. In fact, you could think of the countable numbers as a function that, given a number, produces a new number one higher. However, there is a difference between the "idea" of the function and the numbers that the function produces. The idea of the function can go into the infinite, the actual numbers cannot. The same holds true with PI. The idea of PI can go into the infinite, the actual number can't. The problem with the proof of uncountable infinity is that the natural numbers are limited to the finite whereas the irrational numbers are allowed to go to the infinite. When one realizes that the irrational numbers cannot go to the infinite, then the proof becomes nonsense.
1 comment:
There is another way of considering rational vs. irrational numbers. For example, a number such as 1/7 is infinite in decimal notation, although it recurs, and is a rational number. However, if you chose to work in base 7, it would no longer recur, and would not have infinite length.
All irrational numbers are infinite in length in any numerical base, whereas rational numbers are finite in some base.
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