obsrvr524 wrote:Scientific American wrote:Strange but True: Infinity Comes in Different Sizes

If you were counting on infinity being absolute, your number's up

As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.

Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the "reals." (This set comprises all numbers that can be represented as a decimal, even if that decimal representation is infinite in length. Hence, 27 is a real number, as is π, or 3.14159….)

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"The idea of being 'larger than' was really a breakthrough," says Stanley Burris, professor emeritus of mathematics at the University of Waterloo in Ontario. "You had this basic arithmetic of infinity, but no one had thought of classifying within infinity—it was just kind of a single object before that."

Adds mathematician Joseph Mileti of Dartmouth College: "When I first heard the result and first saw it, it was definitely something that knocked me over. It's one of those results that's short and sweet and really, really surprising."Wkipedia wrote:Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number.[1] It is often denoted by the infinity symbol ∞.

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The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets.[2] The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics

I could present more if those aren't enough. I hate to keep harping on this but I don't think you are going to get anywhere until you face it.

There may be some repetition here. It's a product of me trying to convey my understanding as effectively as I can. My apologise if it's in excess.

It doesn't really matter who said x. What matters is that x be non-paradoxical. I know what Cantor said and I see a paradox in what he said. He also saw this himself. Thus, clearly, what he said was problematic/paradoxical. I think we should solve this in a genuine manner. Not ignore it and settle for some clearly unfulfilling theory (as all other forms of set theory have attempted to do).

Is there something actually infinite? Yes. Call this x. Does the potentially 'infinite' ever become x? No. So why treat them the same when they are clearly different? Which do you deny? x being x? Or the potentially 'infinite' never becoming x? If you deny neither, then you must acknowledge that they are clearly different. I will illustrate their difference more clearly:

Now if there was nothing Infinite (x) (as opposed to the paradoxical idea of something becoming 'infinite'), there would be no solution to Cantor's paradox. There wouldn't even be a potential 'infinity'. But since infinity is that which encompasses all potential infinites as well as finites, we can solve the following: In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number.

For the sake of argument, take Infinity to be the greatest cardinal number. This does not imply an end to the numbers that forever try to get to Infinity (but never do), it just implies that it is greater than all those numbers that try to get (of which the list is endless), but never do. In this fact alone, should we not separate the infinite from the potentially 'infinite'? Again, a potentially 'infinite sequence' does not contain this greatest number. It tries to reach this number, but never does. Can we really say that this number or quantity does not exist when we are trying to count to it? We know there is nothing beyond infinity. A number that cannot be reached even if one counts forever. Again, one does not reach infinity. Infinity just is infinity. It encompasses all numbers that try to reach it. It cannot be reached even if a number sequence goes on forever. This does not mean it does not exist. Nor does it imply a final end point of the endless.

The word infinity implies the potential for an endless set of numbers (of which the number of these sets could be endless). It does not imply the potential for another infinity, or for multiple or different sizes of infinity. It and only It, IS Infinity.

You say Infinity is not a quantity. This seems to be in line with Cantor saying 'there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number.'. Yet, he acknowledged this to be paradoxical. Per the dictates of pure reason, paradoxes are just cases of misunderstanding semantics or lack of reasoning. Call Infinity the greatest cardinal number. Call Infinity the set of all cardinalities...and you will have no paradoxes. Right? If wrong, where is it wrong? How would any paradox arise from it? And if not wrong, why not adopt it?