“Cantor’s Infinity Proof Made Easy”
Cantor’s proof is an elegant and apparently mathematically valid demonstration that arrives at a surprising conclusion. Like Bells’s
Theorem, which asserts the necessity for non-locality of physical events, at least at the quantum level, the conclusion of Cantor’s proof is counterintuitive and seemingly impossible. Bell’s Theorem, however, has been repeatedly tested by a variety of experiments and every properly conducted experiment supports the theorem. Things just cannot be that way but they are.
Cantor’s proof, on the other hand, makes an assumption that is unfounded and clearly invalid. The attempt is made to state that there are “countable” infinities and “uncountable” infinities. If that assumption is accepted, then it does follow that some infinities must be larger than others.
The problem arises with that term “countable infinity”. There is no such thing. The assumption is that if you had enough time you could count all the members of this particular infinite set. How much time would it take? Eternity? That compounds the first error by making another one. Infinity is not a number. Eternity is not a length of time. Some very smart, even gifted, mathematicians have run into trouble trying to handle infinity (among them, Isaac Newton). Infinity is not a number and cannot be treated as a number. The claim that one infinity is somehow larger than another implies that there is a boundary of some kind to the “smaller” infinity. By definition, there is no upper limit to infinity. The use of set theory to illustrate the “proof” is interesting and entertaining but it is ultimately just entertaining gibberish.
The concept of infinity is a difficult one to wrap your thoughts around, and I am not sure anyone really understands it. If you try to work with it mathematically by treating it as a number (the largest number?; the largest number that could be possible?; both ideas are incorrect) you will almost certainly arrive at absurd conclusions. We may say “an infinite number” when what we mean is “a very large number”. The concepts are not the same; they are not even closely related. Any number, even the number of photons in our universe (~400 per cubic centimeter-however many that comes to) is not infinite. Once you begin to treat infinity as a number or as a concept that is somehow bounded you are almost certain to arrive at an erroneous conclusion.