Actual infinity and Cantor's paradox

[Certainly real]

The universe had a beginning

I think you would have a hard time proving that one. The opposite appears to be more provable.

I don't know. It doesn't really matter. If they say the universe is actual infinity, then I see no paradoxes. But if they say otherwise (for example if they say the universe began as a result of the Big Bang), then the universe is not Existence and it is not Actually/Truly Infinite. It exists in Existence as opposed to being Existence.

[obsrvr524]

You can count forever.
You can count endlessly.
You can count infinitely.

Are they semantically the same? The first two sentences are clearly the same. I don't see the third as being the same. Even if you and I are immortal, we cannot count infinitely. We can only count endlessly/forever because we'll never hit infinity to be able to describe ourselves as successfully counting infinitely.

I think you are wrong about that one.

You can count forever - because there is always ever more.
You can count endlessly - because there is no end.
You can count infinitely - because there is no infinity.

The word "infinity" implies the final end point of the endless. It is intended to reference a direction, not an end goal. It is like saying "I counted to up". There is no "to up". There is only "toward up". And there is only "toward infinity" or perhaps "toward the infinite".

None of them have anything to do with someone's ability to count. They each express that the target does not exist except as a direction.

What else could it mean?

[Pedro I Rengel]

I think you are wrong about that one.

You can count forever - because there is always ever more.
You can count endlessly - because there is no end.
You can count infinitely - because there is no infinity.

The word "infinity" implies the final end point of the endless.

Ok you know what? That's actually quality.

[Certainly real]

You can count forever - because there is always ever more.

And this is because there is infinity. You could not count forever if there was no infinity.

You can count infinitely - because there is no infinity

The word "infinity" implies the final end point of the endless.

The 'final end point of endlessness' is contradictory.

It is intended to reference a direction, not an end goal. It is like saying "I counted to up". There is no "to up". There is only "toward up". And there is only "toward infinity" or perhaps "toward the infinite".

None of them have anything to do with someone's ability to count. They each express that the target does not exist except as a direction.

Yes but importantly, it means that the target exists as a direction in the infinite. If there wasn't the infinite, x could not go up forever, count forever, live forever, and so on. No matter what x does, x will never become infinite. Even if it tried to do so forever, it still wouldn't succeed. Only the infinite is infinite and no non-infinite thing can become infinite from a non-infinite state. No non-omnipresent/omnipotent being can become omnipresent/omnipotent. How will it do this? Will the omnipresent go into non-existence and make way for this non-omnipresent thing to take its place? Infinity is literally descriptive of Existence/God. All things are possible or true because of It. All things happen in It, because of It. Being able to count endlessly is a hypothetical possibility because of It. Being Infinite is not a hypothetical possibility because nothing can become infinite. Something just is Infinite.

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. A part of Infinity, is not the same as Infinity.

[Magnus Anderson]

If the word "infinity" means "without an end", then that which has no end is infinite EVEN IF it has a beginning.

You can count forever. You can count without end. But you cannot count to infinity. Infinity is that which you are trying to count to. You can do this forever (note that I did not say "you can do this infinitely'). Compare the following three sentences:

You can count forever.
You can count endlessly.
You can count infinitely.

Are they semantically the same? The first two sentences are clearly the same. I don't see the third as being the same. Even if you and I are immortal, we cannot count infinitely. We can only count endlessly/forever because we'll never hit infinity to be able to describe ourselves as successfully counting infinitely.

I am not sure that counts as a valid response to what I said.

But I'll respond to what you wrote anyways.

"To count infinitely" means "to count without an end". If you start with \(0\) and then move upwards by counting every number greater than \(0\) but less than \(1\), such a process of counting would be infinite in the sense that it would be without an end which really only means there won't be such a thing as "the last number counted". Instead, for every number counted, there will be a number counted at a later point in time. It does not mean the process of counting will never be completed (in theory, it can be completed within any finite period of time one can think of) and it certainly does not mean the last number counted would be \(1\) (or any other number.)

[Meno_]

If the word "infinity" means "without an end", then that which has no end is infinite EVEN IF it has a beginning.

And here , discernability interjects a necessary element,

You can count forever. You can count without end. But you cannot count to infinity. Infinity is that which you are trying to count to. You can do this forever (note that I did not say "you can do this infinitely'). Compare the following three sentences:

You can count forever.
You can count endlessly.
You can count infinitely.

Are they semantically the same? The first two sentences are clearly the same. I don't see the third as being the same. Even if you and I are immortal, we cannot count infinitely. We can only count endlessly/forever because we'll never hit infinity to be able to describe ourselves as successfully counting infinitely.

I am not sure that counts as a valid response to what I said.

But I'll respond to what you wrote anyways.

"To count infinitely" means "to count without an end". If you start with \(0\) and then move upwards by counting every number greater than \(0\) but less than \(1\), such a process of counting would be infinite in the sense that it would be without an end which really only means there won't be such a thing as "the last number counted". Instead, for every number counted, there will be a number counted at a later point in time. It does not mean the process of counting will never be completed (in theory, it can be completed within any finite period of time one can think of) and it certainly does not mean the last number counted would be \(1\) (or any other number.)

This is at the point where discernability introjects a new progression of an unseen and paradoxical element.

This is the point at which the uncertianty principle of checking if the cat in the box is alive or dead pops up.

At this point, the nominal distinction between actual and probable infinity becomes a wash.

[obsrvr524]

The word "infinity" implies the final end point of the endless.

The 'final end point of endlessness' is contradictory.

That was my point. And that is why it can't be counted to.

it means that the target exists as a direction in the infinite.

What does "in the infinite" mean? Is 1 million "in the infinite"? Is infinity "in the infinite"? If it is "in" the infinite then it cannot be the infinite (at least not that same infinite).

If there wasn't the infinite, x could not go up forever, count forever, live forever, and so on. No matter what x does, x will never become infinite. Even if it tried to do so forever, it still wouldn't succeed. Only the infinite is infinite and no non-infinite thing can become infinite from a non-infinite state. No non-omnipresent/omnipotent being can become omnipresent/omnipotent. How will it do this? Will the omnipresent go into non-existence and make way for this non-omnipresent thing to take its place?

I believe you are right in that part. But what I have been saying is that there is no infinity - the end point of the endless. There is no "up" except as a direction - "upward". "Upty" doesn't exist (unless maybe you're English ).

Infinity is literally descriptive of Existence/God. All things are possible or true because of It. All things happen in It, because of It. Being able to count endlessly is a hypothetical possibility because of It. Being Infinite is not a hypothetical possibility because nothing can become infinite. Something just is Infinite.

You should probably read up on James about that (James S Saint not St James). He goes into that issue in extreme extreme detail. You might even say "infinite detail".

There is an infinity. This is the same as saying there are an infinite number of things (this denotes the whole of Existence).

I still disagree.

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.

We should not say "infinite number" because there is no number that is infinite. It has nothing to do with existence. It is a matter of definition of the words. There is an "infinity of numbers". There is not an "infinite number" - a number that is infinite (an oxymoron). And "infinity" implies an infinite number.

[Certainly real]

What does "in the infinite" mean? Is 1 million "in the infinite"? Is infinity "in the infinite"? If it is "in" the infinite then it cannot be the infinite (at least not that same infinite).

Every thing is in the Infinite. Yes, infinity is in the infinite. There is Existence in Existence. There is Infinity in Infinity. Relative to us internally is infinitesimal (which is infinite), and relative to us externally is infinity (which is infinite).

If it is "in" the infinite then it cannot be the infinite (at least not that same infinite).

I think the logic of set theory indicates that it is in the infinite and that it is the same infinite. It's just from which point of view we are looking at it.

I still disagree.

I still think set theory conclusively shows that the set of all sets is that which is infinite (Existence). Existence is an existing thing, and all things that exist within it (including itself) are existing things too. The sum/set of all existing things = infinity. The sum of all books = semi-infinity. It is not infinity because it does not denote or refer to all existing things. This is why cantor's paradox is easily solved when we say recognise infinity as the set of all cardinalities because infinity truly contains all cardinalities. It is because of the infinite that there is always one number more. Thus It is infinity that contains and accommodates the set of all cardinalities.

We should not say "infinite number" because there is no number that is infinite. It has nothing to do with existence. It is a matter of definition of the words. There is an "infinity of numbers". There is not an "infinite number" - a number that is infinite (an oxymoron). And "infinity" implies an infinite number.

We might have to agree to disagree on this. I think Existence exists. It is Actually/Truly infinite. When all of it is highlighted from a purely quantitative perspective, then all things within it and itself are highlighted. Thus the number of existing things = infinity. Infinity is the biggest number. When less than this is highlighted, then less than infinity is highlighted. Semi-Infinity is highlighted.

[obsrvr524]

We might have to agree to disagree on this. I think Existence exists. It is Actually/Truly infinite. When all of it is highlighted from a purely quantitative perspective, then all things within it and itself are highlighted. Thus the number of existing things = infinity. Infinity is the biggest number. When less than this is highlighted, then less than infinity is highlighted. Semi-Infinity is highlighted.

I am often tempted to encourage a dog to distinguish color but I won't endlessly argue with him about it. That applies to several people on this board.

I believe that you are misusing the word "infinity" and you have made it an inseparable part of your arguments concerning paradoxes and God. So your argumentation will be endless - infinite without ever being able to reach an end - futile.

And so I think your bubble of belief will be your own isolated world - affecting no one but keeping you afloat without rise.

None of my argument has been about existence.

[Meno_]

I do agree that Leibnitz is a wishful thinker, but...........what if ..... trying to remember ....zizec's use of bubbles. Is there at least a probable reference that may make some discerning sense for this shared metaphore?

[obsrvr524]

I do agree that Leibnitz is a wishful thinker, but...........what if ..... trying to remember ....zizec's use of bubbles. Is there at least a probable reference that may make some discerning sense for this shared metaphore?

Strange - I was thinking about the yesterday.

I see Zizek as one of those many "probable for suicide" type of philosophers who can't find solutions, only problems to rant about - a bubble that inevitably bursts in the light.

He hates the very water that keeps him alive.

[Meno_]

I do agree that Leibnitz is a wishful thinker, but...........what if ..... trying to remember ....zizec's use of bubbles. Is there at least a probable reference that may make some discerning sense for this shared metaphore?

Strange - I was thinking about the yesterday.

I see Zizek as one of those many "probable for suicide" type of philosophers who can't find solutions, only problems to rant about - a bubble that inevitably bursts in the light.

He hates the very water that keeps him alive.

Typical post modern insecuriry .

[Certainly real]

We might have to agree to disagree on this. I think Existence exists. It is Actually/Truly infinite. When all of it is highlighted from a purely quantitative perspective, then all things within it and itself are highlighted. Thus the number of existing things = infinity. Infinity is the biggest number. When less than this is highlighted, then less than infinity is highlighted. Semi-Infinity is highlighted.

I am often tempted to encourage a dog to distinguish color but I won't endlessly argue with him about it. That applies to several people on this board.

I believe that you are misusing the word "infinity" and you have made it an inseparable part of your arguments concerning paradoxes and God. So your argumentation will be endless - infinite without ever being able to reach an end - futile.
None of my argument has been about existence.

You say I am misusing the word infinity whilst I believe you are misusing it. We have discussed it and I feel like I have said all that I should have said regarding why I use it the way that I do. So I don't think we will agree on the usage of the word infinity. You say none of your arguments have been about Existence. All of my arguments are about Existence. I don't try to make sense of anything (including infinity) independently of Existence, because all truths are true purely as a result of Existence being the way that it is. Existence is the way it is because it just is the way it is. Triangles are three sided because Existence is the way it is. Infinity is the way it is because Existence is the way it is (Infinite).

And so I think your bubble of belief will be your own isolated world - affecting no one but keeping you afloat without rise.

I know everyone gets what they truly/perfectly deserve without fail. If that's what I truly/perfectly deserve, then I think I'm at peace with that. If I deserve less then that, again, I think I'm at peace with that. If I deserve more than that, the same except I'd be happier. I am not omniscient with regards to my own soul. Hopefully, I deserve more than what you describe. I'm hesitant to say I think I deserve more than what you describe because I feel somewhat arrogant saying it. I don't want to be optimistic towards myself at the expense of being arrogant.

[Certainly real]

If the word "infinity" means "without an end", then that which has no end is infinite EVEN IF it has a beginning.

You can count forever. You can count without end. But you cannot count to infinity. Infinity is that which you are trying to count to. You can do this forever (note that I did not say "you can do this infinitely'). Compare the following three sentences:

You can count forever.
You can count endlessly.
You can count infinitely.

Are they semantically the same? The first two sentences are clearly the same. I don't see the third as being the same. Even if you and I are immortal, we cannot count infinitely. We can only count endlessly/forever because we'll never hit infinity to be able to describe ourselves as successfully counting infinitely.

I am not sure that counts as a valid response to what I said.

But I'll respond to what you wrote anyways.

"To count infinitely" means "to count without an end". If you start with \(0\) and then move upwards by counting every number greater than \(0\) but less than \(1\), such a process of counting would be infinite in the sense that it would be without an end which really only means there won't be such a thing as "the last number counted". Instead, for every number counted, there will be a number counted at a later point in time. It does not mean the process of counting will never be completed (in theory, it can be completed within any finite period of time one can think of) and it certainly does not mean the last number counted would be \(1\) (or any other number.)

I missed your post. Sorry for the late reply.

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?

[Certainly real]

I just wanted to correct myself. In the opening post, I said that there are an infinite number of books. I then later said that there aren't an infinite number of books because infinity must denote the whole of Existence and the set of all books does not do this, therefore it should not be called infinite. The opening post was correct. There are in fact an infinite number of books for the following reason:

x contains all books. Let's say R contains all of x except book '10'. Since it is missing book 10, it is classified as incompletely containing x as opposed to completely and infinitely containing x. Where what has been defined is not absurd (it is absurd as we will come to see), this library is the closest thing to infinitely/completely/truly containing x. It matches x 99.99999.....ad infinitum%. There can be no infinite library because no library can 100% contain x or be Infinity/Omnipresent. Where R contains all books except '10' and '9', then R does not match x 99.99999...ad infinitum%. The best that we can describe it is that it matches x 99.99999...ad infinitum% minus one book, or, we can describe it as matching x 100% minus 2 books. R is clearly absurd because infinity - 1 = absurd. It does not = infinity. Nor does it = infinity - 1. You cannot describe a part of Infinity unless you are describing semi-infinites or finites. There is no other class to discuss. There is no in between semantical value. There is no x - 1. There is either a semi-infinite part of x or a finite part of x.

You cannot have two xs because x encompasses all books (both hypothetical or as real as us). You cannot add or take away from x because x is infinite.

There clearly can be different finite and semi-infinite sizes. They are all contained in or members of Existence. Existence clearly is Infinite/Omnipresent and there is only one of it. It does not come in different sizes. It contains Itself because the Infinite contains the Infinitesimal and the Infinitesimal is Infinite. Thus, the Infinite = the Infinitesimal and it contains Itself. This is why x is at all meaningful and why nothing else can contain x other than the Infinite or the Infinitesimal. Just as you cannot reach Infinity by trying to count to Infinity endlessly, you can never expand or shrink to Infinity/Infinitesimal by trying to forever shrink/expand to Infinitesimal/Infinity. Despite this the Infinite contains the Infinitesimal and the Infinitesimal is Infinite. There aren’t multiple infinities or infinitesimals or existences. That is an illusion.

[obsrvr524]

I think you would get stuck on that issue of whether 1 = .999....

If you take 1 away from an infinite count, you still have an infinite count. So are you going to say that an infinite count doesn't go to infinity?

[Meno_]

.

[Certainly real]

I think you would get stuck on that issue of whether 1 = .999....

If you take 1 away from an infinite count, you still have an infinite count. So are you going to say that an infinite count doesn't go to infinity?

In my opinion, taking one away from an infinite count or quantity, does not = infinity. It equals absurdity. An infinite number of books does not go to infinite in quantity, it IS infinite in quantity. I know you are talking about infinity and not existence, but I cannot separate these two concepts from each other. Just as I think it absurd to take 1 book away from Existence, I think it absurd to take 1 book away from an infinite number of books. Which is why I feel the need for the label of semi-infinite to fill in the following semantical gap:

I can have a library that goes on forever containing an endless number of books. But this library can never expand to the point of infinity in the same way that I can never count to infinity. But despite this, it still goes on forever. Infinity is true and meaningful. This is the only reason why counting to infinity forever without reaching infinity is possible. Or why a semi-infinite library can contain an endless/semi-infinite number of books without the number of books it contains reaching infinity. Infinity is also why reaching infinity is impossible. Infinity is not a hypothetical possibility. It just necessarily is infinite and nothing can become it. Infinity reaches all things. It encompasses all things. All that is in it will never see it, reach it, or become it. Of course, this does not mean that we have no understanding of it.

[obsrvr524]

In my opinion, taking one away from an infinite count or quantity, does not = infinity. It equals absurdity.

So the set of all real numbers except 0 is not an infinite set?

I think the world of maths will strongly disagree.

I know you are talking about infinity and not existence, but I cannot separate these two concepts from each other.

So why not just use one of the terms for sake of the rest of us who try to understand what you are saying?

[Certainly real]

So the set of all real numbers except 0 is not an infinite set?

It is. But you cannot take anything away from this set.

So why not just use one of the terms for sake of the rest of us who try to understand what you are saying?

Because maths claims that you can take away from Infinity and be left with Infinity. How have mathematicians come up with this??? You go to someone and ask can you take anything from Infinity? They may not be sure about the answer. You go to someone and ask can you take anything away from Existence? Again, they may not be sure about the answer. Make clear to them the semantical/logical implications of something exiting or entering Existence, and they will say no, you cannot add or take away from Existence. Further make clear to them that by definition, Existence is Actually/Truly Infinite, and then ask them, can you take away from Infinity? They will surely then answer no.

You can take finite books out of a finite library, but you cannot take these books out of x (the set of all books). You can even burn these books and turn them into ashes, but x is still infinite. Those books that you burnt, did not cease to exist in Existence for you to say x -1 occurred. Nor did they exit Existence. The past does not go out of Existence. So those books that were books at that point in time and reality, are still books at that point in time and reality.

We look to the stars, and what we see is their past state. This is because of the speed of light.

[obsrvr524]

So the set of all real numbers except 0 is not an infinite set?

It is. But you cannot take anything away from this set.

I just did, didn't I?

All I am trying to say is that you are using words that means differently to others than what you mean by them. That is leading to others being confused by what you say as well as some logic mistakes you make because logic requires consistency in your words.

But I didn't mean to distract. I have said enough on this - for now anyway.

[Meno_]

Certainly Real says:


"The past does not go out of Existence. So those books that were books at that point in time and reality, are still books at that point in time and reality.

We look to the stars, and what we see is their past state. This is because of the speed of light......"


Existence has no being as a foundation, and Being as a foundation negates Russell's idea in the philosophycal foundations as logico-mathematical, and destroys his ' sense-data idea by incrementally shortening out the overbearing data, by enscription.

In other words, the symbolic structure creates memory gaps, as it has to generalize programs of relevance, but the epigrams do not somehow 'leak' that information

Therefore the reductive process of stored data can not be a foundation in. ' real' sense, when sensation in the instant can not connect the sufficient number of frequency of those illusive stills, that are at the very bottom , or required repetition of assimilated epigrams.

The eye can only capture a still and it is by the means of internal simulation , that those tiny 'beings' can coalesce into motion.

Cantor can only imagine an all inclusive set, but he can actually quantify an infinite series where every set underneath , not unlike an infinite set of turtles, are one on top of another

In this instance, you can imagine a calculus of fed back epigrams, which creates the illusion of flow..

That is where the difference between the two differentials on both sides of a gap, or a memory loss, due to symbolic generalization.

This IDEA formation harbors a push for learning( conscious), because projective (or objective) ideas can not create anything but illusions of movement.

A retrograde process of forming constant , epigram 'gaping' is the result of a universal move , toward a all inclusive foundation,where 'gravity' is the unmoved moved.

It is a set that can , or can not include It's self at
in that instant. I think somewhere along described it as 'eigenblick'

That is not God, but it is a power that enables it to connect a the gaps between what slips out of these gaps..So God becomes that power, which includes as own source.( of power)



ref: K .Polanyi:The Great Transformation

"Embeddedness” centers around Karl Polanyi's concept of ... from a set of social mechanisms that resolve underlying problems.




"Understanding, comprehension – this is the cognitive faculty cast aside by a positivistic theory of knowledge, which refuses to acknowledge the existence of comprehensive entities as distinct from their particulars; and this is the faculty which I recognize as the central act of knowing. For comprehension can never be absent from any process of knowing and is indeed the ultimate sanction of any such act. What is not understood cannot be said to be known. (p. 240)

Like many other philosophers I enjoy, Polanyi stresses the movement of a knowing mind rather than any static state of knowledge.

The dynamic impulse by which we acquire understanding is only reduced and never lost when we hold knowledge acquired and established by this impulse. The same impulse sustains our conviction for dwelling in this knowledge and for developing our thoughts within its framework. Live knowledge is a perpetual source of new surmises, an inexhaustible mine of still hidden implications. (p. 244)"

[Certainly real]

Right, I think I've reached a definitive decision on this.

Call the set of all lions x. Call the set of all animals y. Neither x nor y can be Infinite in quantity for the following reason:

It cannot be denied that y is greater in quantity than x. This is because y encompasses the quantity of x. Thus, it is absurd to say x and y are equal in terms of quantity when y wholly encompasses x's quantity. There are more things than just lions and animals. A greater quantity than that which is representative of the quantity of animals exists. A greater number/quantity exists than the quantities that x and y denote.

There is no quantity greater than Infinity. What is there an Infinite number/quantity of? Things. There are an Infinite number of things. The semantic of thing encompasses all things (this includes numbers). There are more things than there are animals. There are no more things than there are things. Thus, Infinity is only representative of the total number of things, or the set of all things. Not the total number of lions or the set of all animals.

Thus, you cannot have an infinite number of lions or animals. You have an infinite number of things.

It is true that you can have an endless number of animals and lions. But it is also true that there are more animals than there are lions. Thus, semi-infinite is an appropriate word to use here to describe that which is not Infinite in quantity, but still endless and can come in varying sizes. You can have different sizes of finite and semi-infinite. But you cannot have different sizes of Infinity. That implies that you can have different size of the greatest quantity. Or that you can have different sizes of that which no greater quantity can be conceived of.

If x is infinite and y is infinite yet y is larger in quantity than x, then x is certainly not infinite. y is only infinite when there is nothing larger in quantity than it.

[obsrvr524]

There is no quantity greater than Infinity.

Infinity is NOT a quantity.

If x is infinite and y is infinite yet y is larger in quantity than x, then x is certainly not infinite. y is only infinite when there is nothing larger in quantity than it.[/color]

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.

[Magnus Anderson]

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.