Actual infinity and Cantor's paradox

[Certainly real]

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….)
~~~
"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."

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 ∞.
~~~
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?

[Certainly real]

I think I understand where you are coming from. To better understand my position, how would you respond to the following:

There is an infinity. This is the same as saying there are an infinite number of things (this denotes the whole of Existence). We should not say there is an infinite number of books because an infinite number of books, does not denote the whole of Existence/Infinity. It denotes a part of it because books are not all there is in relation to Existence. A part of Infinity, is not the same as Infinity. If we say there is an infinite number of books, we are only highlighting a part of that which is Actually/Truly Infinite. If we're only highlighting a part of it, is it not paradoxical to say that we have highlighted all of it? Isn't it better to label a finite part of infinity as finite? Or to label an endless part of Infinity as semi-infinite?

I agree that "infinity" means "an infinite number of things". However, I do not agree that "an infinite number of things" means the same as "the whole of existence". In other words, if you say that the something is infinite in size, it does not mean that its size is equal to the size of the universe (i.e. that it occupies every bit of physical space.) The word "infinite" simply means "a number greater than every integer" (that's the definition I prefer since it's the clearest one.) By definition, it is unrelated to what exists.

For me, Existence and the universe are not the same. I think there is more to Existence than just our universe. I think this for the following reason: The universe had a beginning. It is paradoxical for something to come from nothing, or to have had a beginning in non-existence/nothingness. To avoid the paradox of something coming from nothing, we need an entity/being/thing/Universe, that is actually infinite. I call this thing Existence. If Existence was temporally finite, then that implies that it came from non-existence. If it was spatially finite, then that implies it is surrounded by non-existence. But then that paradoxically implies that non-existence exists. For how can non-existence surround Existence if it does not exist? In other words, how can x be surrounded by that which does not exist? It cannot.

Paradoxes are all the same. They must not be accepted on any level. We do not accept triangles being four-sided. We do not accept x being in two different places at the same time. We do not accept something coming from nothing. Given the paradoxes in the aforementioned paragraph, I see myself as rationally obliged to acknowledge Existence as being actually/truly Infinite.

[promethean75]

"The universe had a beginning."

you'ont know that, bro.

[Certainly real]

"The universe had a beginning."

you'ont know that, bro.

True. If it did, then it does not represent the whole of Existence.

[obsrvr524]

For the sake of argument, take Infinity to be the greatest cardinal number. [color=#004080]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.

You seem to have it stuck in your head that infinity is an actual thing. It isn't. It is an oxymoron intended to just imply the "direction toward the infinite". Infinity itself does exist any more than "up" exists as an actual height. It is nonsense to keep treating infinity as though it was actual existence and then compare it to real existence.

And who said anything about "becoming infinite"? We have all agreed that there is no "becoming infinite". Why are you even bringing it up?

"Potentially infinite" can ONLY mean - "something that might be infinite but we don't know yet". It has nothing to do with growing to become infinite.

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?

There is no "trying to reach". The issue is that the very definition of "infinite" already means that there is no end to be reached whether anything is trying or not - no such thing as "infinity" - a "square-circle" - the "end 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.

No. That is NOT what it means in English. If you want to make up your own language please try to use different words than English.

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?

Wrong. You are claiming that "infinity is the universe" and at the same time claiming that a nonexistent concept, an oxymoron, "infinity" IS that universe that clearly exists.

That would definitely be a paradox as you ignore your word confusions.

You seem to have this list of discoordinate liberal concepts and their words that refute logic and rationality. With your words, you are saying that the universe itself and all reality is a square-circle - balderdash.

[Magnus Anderson]

For me, Existence and the universe are not the same. I think there is more to Existence than just our universe.

But for others, the two words are almost synonymous with each other. The universe is merely the set of all that has existed, that exists and that will exist. Existence is merely whatever exists.

If it was spatially finite, then that implies it is surrounded by non-existence.

Doesn't that mean it is surrounded by nothing? What exactly is the problem with that idea?

But then that paradoxically implies that non-existence exists.

How? Not that it's relevant. It doesn't appear related to Cantor and infinity but . . . just wondering.

[Certainly real]

You seem to have it stuck in your head that infinity is an actual thing.

For the sake of argument, I will try your approach and say there is no actually infinite thing.

"Potentially infinite" can ONLY mean - "something that might be infinite but we don't know yet". It has nothing to do with growing to become infinite.

It is nonsense to keep treating infinity as though it was actual existence and then compare it to real existence.

Given that you say there is no actually infinite thing, then it follows that you don't believe the universe to be an infinite thing. Did the universe have a beginning? Did it come from nothing? Is the universe surrounded by nothing/non-existence (as in is it spatially finite)?

And who said anything about "becoming infinite"? We have all agreed that there is no "becoming infinite". Why are you even bringing it up?

To make a distinction between that which has no end, but has a beginning, and that which has no beginning and no end. x can go on forever, but because it had a beginning, it will never become that which has no beginning and no end (I've refrained from using the word infinity here, though I think it necessary to have a separate label for that which has no beginning and no end, versus that which has a beginning but no end)

There is no "trying to reach". The issue is that the very definition of "infinite" already means that there is no end to be reached whether anything is trying or not - no such thing as "infinity" - a "square-circle" - the "end of the endless".

Ok, I will not say trying to reach infinity. I will say, it goes on forever, but because it had a beginning, it is distinct from that which had no beginning and no end (which I will refrain from calling Infinity).

No. That is NOT what it means in English. If you want to make up your own language please try to use different words than English.

Ok, let me know what your take is on the following:

Is the universe such that it had no beginning and has no end? If no, do you consider there to be another existing thing x that encompasses the universe such that x had no beginning and has no end?

Wrong. You are claiming that "infinity is the universe" and at the same time claiming that a nonexistent concept, an oxymoron, "infinity" IS that universe that clearly exists.

Personally, I don't view the universe as infinite. I think the universe had a beginning, therefore, I don't think it is infinite.

That would definitely be a paradox as you ignore your word confusions.

You seem to have this list of discoordinate liberal concepts and their words that refute logic and rationality. With your words, you are saying that the universe itself and all reality is a square-circle - balderdash.

I'm keen to find out what your view on reality is. I look forward to your answers to the questions I've asked.

[Certainly real]

[
Doesn't that mean it is surrounded by nothing? What exactly is the problem with that idea?

If x is surrounded by y, then that logically implies that y exists. If y does not exist, then y does not surround x (because it does not exist to surround x). If the universe is surround by nothing, then that logically implies that nothing exists. Is it not paradoxical to say that nothing/nothingness or non-existence exists?

How? Not that it's relevant. It doesn't appear related to Cantor and infinity but . . . just wondering.

First I have to make clear that something that has no beginning and no end (which I would label as actually/truly Infinite) exists and that it is paradoxical to reject this. Once I do this, I can then show that that which is actually/truly Infinite, encompasses all things that have a beginning, but no end. I can then make a distinction between this thing which is actually Infinite, and the things that have a beginning but no end (which I would label semi-infinite). I can then show that Cantor's paradox can be solved by acknowledging that that which has no beginning and no end, encompasses all things that have a beginning but no end (which mathematicians are calling infinite). Since Infinity encompasses semi-infinities, we have a set that contains the set of all semi-infinities, whilst it Itself, is Infinite.

[Magnus Anderson]

If x is surrounded by y, then that logically implies that y exists. If y does not exist, then y does not surround x (because it does not exist to surround x). If the universe is surround by nothing, then that logically implies that nothing exists. Is it not paradoxical to say that nothing/nothingness or non-existence exists?

It is paradoxical to say that non-existence exists. That's not what I dispute. What I dispute is what the statement "X is surrounded by nothing" implies. This has to do with what that statement means. I interpret that statement the same way I interpret "X is not surrounded by anything" and I simply don't see how that statement implies that non-existence exists.

[obsrvr524]

You seem to have it stuck in your head that infinity is an actual thing.

For the sake of argument, I will try your approach and say there is no actually infinite thing.

That is NOT what I said.

I said "infinity" as a actual thing or location does not exist. Infinity is an oxymoron of concepts - a bogus word - a square-circle.

Of course there are things that are infinite in scope, such as an "infinite line" or total number of stars or total number of natural numbers or the entire universe.

[Certainly real]

That is NOT what I said.

You said "You seem to have it stuck in your head that infinity is an actual thing. It isn't." So I said "For the sake of argument, I will try your approach and say there is no actually infinite thing." It may not be what you meant. But it is what you said.

I said "infinity" as a actual thing or location does not exist. Infinity is an oxymoron of concepts - a bogus word - a square-circle.

Right, so let's discuss this point then. So you said:

Of course there are things that are infinite in scope, such as an "infinite line" or total number of stars or total number of natural numbers or the entire universe.

I take this as you meaning that the entire universe is infinite. If it's infinite in scope, then it's infinite. It cannot be infinite in scope and yet not infinite at the same time. So:

Is the universe such that it had no beginning and has no end on any level, degree or dimension? (by this I mean it truly has no bounds/limits in any aspect or dimension. For example, a room that has a finite length and height, but is endless in terms of width in both directions, is still limited in length and height, thus it is not truly unlimited)

If you don't think the universe is this way, then how do you account for the paradox of something coming from nothing? If you do think the universe is this way, can you give me one other thing that also has no beginning and no end on any level, degree, or dimension?

[Certainly real]

I interpret "X is not surrounded by anything"

Right. And if X is not surrounded by anything, then X is Infinite is it not? Consider the following:

X is Infinite, therefore, there is nothing beyond it or outside of it or surrounding it or encompassing it because nothing can encompass actual Infinity.
Y is finite, therefore, there is something beyond it or outside of it or surrounding it or encompassing it because something can encompass finite things. Infinity is one such thing, as are all bigger finite things and semi-infinite things.

[obsrvr524]

That is NOT what I said.

You said "You seem to have it stuck in your head that infinity is an actual thing. It isn't." So I said "For the sake of argument, I will try your approach and say there is no actually infinite thing." It may not be what you meant. But it is what you said.

I said "infinity" as a actual thing or location does not exist. Infinity is an oxymoron of concepts - a bogus word - a square-circle.

You are still not understanding what I actually said. You conflate "infinity" with "infinite". Those are two different concepts. The former doesn't exist as a valid concept. It is a "square-circle" - "the end point of endlessness". The latter is merely another word for "endless".

Is the universe such that it had no beginning and has no end on any level, degree or dimension? (by this I mean it truly has no bounds/limits in any aspect or dimension. For example, a room that has a finite length and height, but is endless in terms of width in both directions, is still limited in length and height, thus it is not truly unlimited)

As stated I don't think that is true. The universe has limits on certain issues - only 3-dimensions, speed of light, many more.

If you don't think the universe is this way, then how do you account for the paradox of something coming from nothing?

I think it is a simple matter of definition that something cannot come from nothing. If something came from t, it wasn't nothing. Nothing = that which has no affect.

If you do think the universe is this way, can you give me one other thing that also has no beginning and no end on any level, degree, or dimension?

Your imagination.

[Certainly real]

As stated I don't think that is true. The universe has limits on certain issues - only 3-dimensions, speed of light, many more.

Being only 3 dimensional does not make a thing limited when there is nothing beyond 3-dimensionality. Also, how does the speed of light being limited = the universe being limited?

Your imagination.

Your imagination had a beginning. Thus it is limited in this aspect is it not?

Which is limited in some way and which is not limited in any way: That which has no beginning and no end, or, that which has a beginning but no end?

Or are they both equally unlimited/infinite?

You are still not understanding what I actually said. You conflate "infinity" with "infinite". Those are two different concepts. The former doesn't exist as a valid concept. It is a "square-circle" - "the end point of endlessness". The latter is merely another word for "endless".

Ok, I will not call that which is actually infinite 'infinity' in our discussion anymore.

[obsrvr524]

Being only 3 dimensional does not make a thing limited when there is nothing beyond 3-dimensionality.

That seems a silly sort of thing to say.
There is no limit - as long as you don't go over 3.

Also, how does the speed of light being limited = the universe being limited?

The universe can't produce anything faster. It is limited by the speed of light.

Ok, I will not call that which is actually infinite 'infinity' in our discussion anymore.

OK. That's good. Only a few more to go.

[Certainly real]

That seems a silly sort of thing to say.
There is no limit - as long as you don't go over 3.

If I said to a child you can have an unlimited number of cookies so long as it's not more than 3, then I am of course being silly and contradictory. If I say to you "Being only 3 dimensional does not make a thing limited" when there is something beyond 3-dimensionality, then again, I am being silly and contradictory. But if there is nothing beyond 3-dimensionality and I express this, then I would not describe me as saying something silly.

You cannot meaningfully describe that which is Omnipresent as being limited, when you cannot meaningfully conceive of how it can be less limited can you? Can you conceive of 4-dimensionality? Does "Being only 3 dimensional does not make a thing limited when there is nothing beyond 3-dimensionality." still seem silly to you?

The universe can't produce anything faster. It is limited by the speed of light.

The universe also can't produce a round square or a gsjiofgj, yet I would not call the universe as being limited because of this. To show something is limited, you have to meaningfully show this. You cannot present absurdities (faster than light travel) or unknowns (more than 4-dimensionality) as meaningful objections or responses to an argument. If you think it paradoxical for something to be faster than the speed of light, then you do not describe the inability to produce something like this as being a limit on the thing that is unable to produce it.

x is meaningfully doable. y can't do x. y is therefore limited.
z is hypothetically impossible and therefore meaningless. y can't do z. It doesn't matter because z is meaningless.

[obsrvr524]

That seems a silly sort of thing to say.
There is no limit - as long as you don't go over 3.

If I said to a child you can have an unlimited number of cookies so long as it's not more than 3, then I am of course being silly and contradictory. If I say to you "Being only 3 dimensional does not make a thing limited" when there is something beyond 3-dimensionality, then again, I am being silly and contradictory. But if there is nothing beyond 3-dimensionality and I express this, then I would not describe me as saying something silly.

You cannot meaningfully describe that which is Omnipresent as being limited, when you cannot meaningfully conceive of how it can be less limited can you? Can you conceive of 4-dimensionality? Does "Being only 3 dimensional does not make a thing limited when there is nothing beyond 3-dimensionality." still seem silly to you?

The universe can't produce anything faster. It is limited by the speed of light.

The universe also can't produce a round square or a gsjiofgj, yet I would not call the universe as being limited because of this. To show something is limited, you have to meaningfully show this. You cannot present absurdities (faster than light travel) or unknowns (more than 4-dimensionality) as meaningful objections or responses to an argument. If you think it paradoxical for something to be faster than the speed of light, then you do not describe the inability to produce something like this as being a limit on the thing that is unable to produce it.

x is meaningfully doable. y can't do x. y is therefore limited.
z is hypothetically impossible and therefore meaningless. y can't do z. It doesn't matter because z is meaningless.

I'm guess that I have to see that as nonsense on both counts.

[Magnus Anderson]

And if X is not surrounded by anything, then X is Infinite is it not?

I don't think so. I think it merely states that there's nothing outside of X without saying anything about the size of X.

[Certainly real]

That seems a silly sort of thing to say.
There is no limit - as long as you don't go over 3.

If I said to a child you can have an unlimited number of cookies so long as it's not more than 3, then I am of course being silly and contradictory. If I say to you "Being only 3 dimensional does not make a thing limited" when there is something beyond 3-dimensionality, then again, I am being silly and contradictory. But if there is nothing beyond 3-dimensionality and I express this, then I would not describe me as saying something silly.

You cannot meaningfully describe that which is Omnipresent as being limited, when you cannot meaningfully conceive of how it can be less limited can you? Can you conceive of 4-dimensionality? Does "Being only 3 dimensional does not make a thing limited when there is nothing beyond 3-dimensionality." still seem silly to you?

The universe can't produce anything faster. It is limited by the speed of light.

The universe also can't produce a round square or a gsjiofgj, yet I would not call the universe as being limited because of this. To show something is limited, you have to meaningfully show this. You cannot present absurdities (faster than light travel) or unknowns (more than 4-dimensionality) as meaningful objections or responses to an argument. If you think it paradoxical for something to be faster than the speed of light, then you do not describe the inability to produce something like this as being a limit on the thing that is unable to produce it.

x is meaningfully doable. y can't do x. y is therefore limited.
z is hypothetically impossible and therefore meaningless. y can't do z. It doesn't matter because z is meaningless.

I'm guess that I have to see that as nonsense on both counts.

I'm not clear on what you're referring to. Do you acknowledge that the limits that you described the universe as having were not actual limits?

[Certainly real]

And if X is not surrounded by anything, then X is Infinite is it not?

I don't think so. I think it merely states that there's nothing outside of X without saying anything about the size of X.

Suppose you reach the very end of X, the border of X. Nothing is in front of you. Can you go into this nothingness? Or is there some wall that prohibits this? A wall that separates the existing universe from its inner side, and the existing non-existence/nothingness from its outside?

[Magnus Anderson]

Suppose you reach the very end of X, the border of X. Nothing is in front of you. Can you go into this nothingness? Or is there some wall that prohibits this? A wall that separates the existing universe from its inner side, and the existing non-existence/nothingness from its outside?

Here's how I think.

Suppose that the room I currently sit in is the entire universe meaning there's nothing outside of it. What does that mean? Does it mean there's "existing non-existence" outside of the room? I wouldn't say so. I would say it merely means that no matter how hard I tried, I would never succeed in leaving the room and observing what's outside of it. Indeed, regardless of who was in the room, and regardless of what they tried, they would never manage to step outside of it.

[obsrvr524]

Do you acknowledge that the limits that you described the universe as having were not actual limits?

Not at all. I don't see how you could think it makes sense that the universe has only 3 of something yet has no limit on that thing. It is like you are defining "no limit" or "infinite" as "whatever the universe has" on any concern or measurement. It just seems ridiculous.

Political liberals do that but we all accept that they are a bit brainless hypocrites.

[Certainly real]

Do you acknowledge that the limits that you described the universe as having were not actual limits?

Not at all. I don't see how you could think it makes sense that the universe has only 3 of something yet has no limit on that thing. It is like you are defining "no limit" or "infinite" as "whatever the universe has" on any concern or measurement. It just seems ridiculous.

Political liberals do that but we all accept that they are a bit brainless hypocrites.

I don't think you've focused on the point I'm trying to make. This is a yes or no question:
For you to be able to conceive of something as being limited, you must be able to conceive of a less limited version of it, do you not?


With that being said, have you conceived of a 4-dimensional universe?

Political liberals do that but we all accept that they are a bit brainless hypocrites.

I don't think you can speak for all people. You are not omniscient.

[Certainly real]

Suppose you reach the very end of X, the border of X. Nothing is in front of you. Can you go into this nothingness? Or is there some wall that prohibits this? A wall that separates the existing universe from its inner side, and the existing non-existence/nothingness from its outside?

Here's how I think.

Suppose that the room I currently sit in is the entire universe meaning there's nothing outside of it. What does that mean? Does it mean there's "existing non-existence" outside of the room? I wouldn't say so. I would say it merely means that no matter how hard I tried, I would never succeed in leaving the room and observing what's outside of it. Indeed, regardless of who was in the room, and regardless of what they tried, they would never manage to step outside of it.

Ok, I think we'll have to agree to disagree on this.

[obsrvr524]

I think the point is that there is no such thing as "outside the universe" especially since we agree that the universe is 3D infinite.