Actual infinity and Cantor's paradox

Let’s start with what labels I attach to what semantics (the definitions of words I work with):

Infinity = that which has no beginning and no end both internally and externally (aka actual infinity)
Semi-infinite = that which has a beginning but no end (aka potential infinity…what mathematicians appear to be calling infinite, despite it not being infinite)

Call the set of all books x. There exists an infinite number of books. Thus x is infinite but it is not Infinity. A finite library can only finitely contain x. A semi-infinite library can only semi-infinitely contain x. That which is actually Infinite (call this Existence/Infinity) fully/completely/infinitely contains x.

Inifnity contains an infinite number of books. If only Existence/Infinity is Infinite (which It is because there can only be one actually Infinite or Omnipresent being), then only Existence fully contains x. This is a logical/semantical/definitional requirement. A semi-infinite library cannot contain an infinite number of books because by definition, it is semi-infinite (semi-omnipresent at best). Thus, a semi-infinite library cannot fully contain x. Keeping in mind that everything that exists is a member of Existence, consider the following:

x = all books in Existence (the number of which is infinite)

Finite library y contains 1 million books. Thus, y contains a part of x. That part is finite. x is infinite. y does not contain x. It contains a finite part (call this f) of x. It contains fx. There can be an infinite number of fxs. By this I mean there can be an infinite number of finite libraries. All of which, are of course, a part of Existence (they may not be as real as you and me, but they are a part of Existence. Dreams, items of thought, and hypothetical possibilities, don’t go in and of Existence because that would be paradoxical. Some attain reality, some are reflected on, some are experienced, and so on. They do not go out of Existence, nor do they come into Existence)

Semi-infinite library z contains a semi-infinite number of books. Thus, z contains a part of x. That part is semi-infinite. x is infinite. z does not contain x. It contains
a semi-infinite part (call this si) of x. Call this si of x, six. There can be an infinite number of sixs. One library can contain an endless number of red books, but it cannot contain all red books (let alone all books). This will become clearer.

Further understanding the semi-infinite:

P and Q are two identical semi-infinite libraries. We take one book away from P:
P does not have book ‘10’ despite containing a semi-infinite number of other books.
Q contains all the books in P as well as book ‘10’. Before taking out book ’10’, P and Q were identical in the number of books they contained. After taking out book ’10’, it can no longer be said that P and Q are identical in the number of books they contain. I will prove this:

Nothing can be taken in and out of Existence/Infinity. But things can be taken in and out of semi-infinite libraries. Neither libraries contain all books because neither libraries are Infinity/Existence/Omnipresent. Despite this, the libraries are connected to the Infinite. Thus, they can have access to an infinite number of books (like we can have access to an infinite number of imaginative thoughts). Libraries can have access to x, but they cannot contain x. Keep in mind, all libraries (finite or semi-infinite) are members of Existence (the set of all sets).

A and B are identical semi-infinite libraries. Where all the books in A and B are red and blue, and then all the blue books are taken out of A, then A and B are 50% identical and B is twice the size of A. Both libraries still contain a semi-infinite number of books. The semi-infinite number of books in B is more potent/concentrated, or with greater depth and breadth (just like the finite size of a 65 inch flat screen tv is greater than that of a 32 inch flat screen tv. Finite things have dimensions, so do semi-infinite things) than A. Thus, B is greater in semi-infiniteness than A. Just as you can have two finite things be of different sizes, you can have two semi-infinite things be of different sizes too. Consider the following:

Semi-infinite library C has an endless number of copies of the Bible. It contains no other books.

If a finite set of books is added to this semi-infinite library, then this library will increase in the total number of books it contains but the semi-infinite number of books it has will not increase. The increase in its number of books must be expressed as semi-inf x + finite x. If another semi-infinite set of books is added to this library (let’s say an endless number of copies of the Quran), then the semi-infinite number of books will have increased such that the semi-infinite number of books in the library is now twice as big. A more comprehensive or bigger semi-infinite you might say.

The biggest possible semi-infinite library R is defined as follows: That which is closest to literally containing all of x, is the biggest library.

Let’s say R contains all books duplicated and original, except ‘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, 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. Admittedly R seems absurd. You cannot take away from the infinite. You can describe parts of it. What is the biggest part of it that you can describe that is not the whole of it?

You cannot have two xs because x encompasses all books (both original and duplicated). Where R is literally x minus one book, you cannot have more than one R. Existence is such that it contains x and R. R contains almost all of x (which means Existence also contains almost all of x because R is in Existence) but Existence contains one more book than R. Thus, Existence contains x, and the manner of its containing x is such that all but one book is in R (which is also in Existence, thus it is in Existence) and one book is not in R (this one book is also in Existence, thus it is in Existence). Both x and R are members of Existence and not members of themselves. Existence is a member of Itself.

Where R is not absurd, it should be described as a super semi-infinite. I’m almost certain (though not yet fully certain) that there can be no super semi-infinites. But there clearly can be different semi-infinite sizes, and Existence clearly is Infinite/Omnipresent.

Solution to Cantor’s paradox:
E = the set of all sets. E is a member of itself. This is literally describing Actual infinity. This is literally describing Existence. True/actual infinity contains infinitesimal. Infinity and infinitesimal refer to the exact same thing just looked at from a different angle/perspective. Existence is infinite through and through. In relation to us, infinitesimal is the internal aspect of Existence and infinity is the external aspect of Existence. We zoom into ourselves past atomic level, there will be something else ad infinitum (rejecting this implies that Existence has an internal end…which is clearly absurd). We zoom out of ourselves past planet level, there will be something else ad infinitum (rejecting this implies that Existence has an external end…which is clearly absurd). An Existence that is not actually/truly infinite is blatantly absurd as it implies the existence of non-existence or something coming from nothing (which is rooted in the absurdity of the existence of non-existence). A temporally finite existence implies that non-existence existed (absurd) and existence came from it. A spatially finite existence implies that it is surrounded by non-existence. This is absurd as it logically implies that non-existence exists and it is surrounding existence. For us to zoom in to ourselves and find and end (the rejection of infinitesimal) logically implies the rejection of Actually infinity. This is clearly absurd.

A clear distinction must be made between actual infinity and potential infinity. Potential infinity is not truly infinite like actual infinity. So it should not be described as being infinite at all. Actually infinity makes the potentially ‘infinite’ possible. Cantor appeared to have thought that everything is just potential ‘infinity’. He did not seem to recognise or understand true/actual infinity. If he did, he would not say there is no set of all cardinalities. Clearly, the set of all cardinalities is Existence/Actual infinity. Again, actual infinity makes potential ‘infinity’ possible. How clearer can it get in terms of what contains the set of all cardinalities?

What contains infinity? Infinity (and there are no different sizes of infinities)
What contains semi-infinities? Infinity (and there are different sizes of semi-infinities)

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With regard to my own thinking on infinity, something/items are either infinite or they are not… though I agree on the concept of potential infinity, so either it becomes infinite or remains finite, but it cannot be both… we can entertain the idea of otherwise, but the reality is not both.

But potential infinity can never become infinite. Take Existence to be actually infinite. Consider a room in Existence that forever expands. This room will never occupy the whole of Existence. This room will never hit infinity. Just as you cannot count to infinity, you cannot expand to infinity. You can have a library that never ends (a semi-infinite library) but this library is not infinite. You can count forever and your counting will never end. But you will never reach infinity. Such is the nature of Infinity/Existence. It reaches all things, yet no thing reaches it. It sustains all things, yet no things sustain it.

I do not believe that objects and items can be infinite, and so agree with your above assertion, of which I have previously voiced in the last previous thread on Infinity, in here. So no, there cannot be infinite apples in concrete reality, but only in thought and notion.

We cannot count to infinity, but we know that numbers can go on forever, because that is the quality of numbers…they have the capacity to never end, we just haven’t got the capacity to achieve that feat… i.e. we do not live forever.

That’s interestingly put… it then sounds like the infinite is self-sustaining, and the finite, dependent.

Numbers are infinite… numbers (seem to) define the universe… perhaps that is why the universe is then said to be infinite, or imagined as so?

Dp…

I have to disagree with how you use some words and terms and that affects most of your arguments. Infinite is not a size. It is not a quantity. It is not a number.

Many people have argued for different “sizes” of infinite. James calls them “different degrees of infinite” because infinity isn’t an quantity. Infinite is a quality.

The word “infinite” means endless. That is a quality or feature of a list. If you take only half of an endless list, you still have an endless list. But that says nothing about the quantity involved or the “size”. So James called it a difference in degree of endlessness. Half of an endless list is half the degree of endlessness yet is still endless. Half of an infinite thing is still infinite yet it is also still only half.

If I have a set A containing all of the locations along an infinite line, I have an infinite set. The set A is endless.

And if I add to that set A all of the locations along a different infinite line (set B), I again have an endless list but it obviously has more locations involved. I have all of set A and also all of set B. So the combination of both sets is of a greater degree of endlessness than either set alone even though all are endless.

Cantor argued that there are more numbers than in the real set of numbers. He provided a famous matrix to prove his point by it providing numbers that are not already included in the real set. James argued that what he added to the real numbers are not actually numbers but ratios that yield endless decimals. That is only an issue of how you define a “number”. James argued that if it ends with “…” it is not a number but merely an expression that can’t be written in decimal form. “0.999…” is an example.

Cantor also argued that there are cardinals of infinity (similar to James’ “degree”). That would be necessary to include all of the locations within a plane versus all of the points along an x axis. If I understand it correctly cardinals only deal with exponents of an infinite set. James was more basic allowing for simple maths such as infA+1 (the set of whole number locations along an x axis plus 1 location above the axis).

So speaking of “infinity” is insufficient language - which infinite are you speaking of? Some infinite sets include MORE items than other infinite sets (greater “size”).

With all of that in mind, I can’t fully agree with those definitions. “Infinite” merely means endless. It doesn’t matter whether it involves a beginning or an end. So your “semi-infinite” is saying “semi-endless”. If anything, that would only imply a difference in degree of infinite from some other chosen infinite (or a difference in “size” for some people).

And the word “infinity” isn’t a rigorous concept. There is no infinity anywhere. And it doesn’t “contain” anything. Existence is infinite (having infA^3 locations) and contains all real/existing things.

Then also what is a “potential infinite” if it cannot produce an actual infinite? How is anything a potential for something if the something is impossible?

And what does “no end both internally or externally” mean? Internal to what?

Why should we believe that? I can burn a page out of a book. It no longer exists. It’s ashes are not “a page in a book”. That page in a book is “out of existence”.

Nothing is also a member of itself. The set of all sets cannot itself contain the set of all sets (as explained before).

Ok now I get the “internal” versus “external”. But doesn’t “infinitesimal” just mean infinitely small rather than infinitely large?

Unless you are a fan of quantum mechanics.

Unless you are a fan of the big bang theory.

Can you prove that non-existence cannot be a state or situation? James would agree with you but he had a proof.

I think Cantor was saying the same as I have been saying - that a set containing itself as a member yields an impossibly large degree of infinity. Cardinality involves that same issue of having no describable upper limit - a senselessly exponentially expanding multiplication of infinities that exceeds all reality - impossible and not a definable set.

Math seems to imply otherwise. Why can’t there be an infinity of planets throughout the infinite universe with an infinity of apples?

When you say “There’s an infinite number of people standing in front of me” doesn’t that imply that the number of people standing in front of you is greater than every integer?

If so, what is the number of numbers greater than every integer? Zero, one or more than one?

If it’s zero, then the concept of infinity is contradictory (and so one would have to explain the purpose of its existence.)

If it’s one, then infinity actually refers to the largest number i.e. to the number greater than every other number.

If it’s more than one, then there can be larger and smaller infinities (one condition would be that the word “infinite” does not specifically refer to some number greater than every integer.)

The greatest integer plus 0.5.

IKEA chairs aren’t infinite, though they seem to produce them ad-infinitely, lol… can the produced/manufactured/grown be infinite, just because they are contained within an (allegedly) infinite space?

Different degrees of infinity, aren’t then really infinity, so perhaps they need another category to define them… though that may have been already mentioned.

The thing about infinity…

ANY finite number (no matter how large it is) is only an infinitesimal compared to infinity… it’s a nothing. A mere blip, if that.

“A clear distinction must be made between actual infinity and potential infinity. Potential infinity is not truly infinite like actual infinity. So it should not be described as being infinite at all. Actually infinity makes the potentially ‘infinite’ possible. Cantor appeared to have thought that everything is just potential ‘infinity’. He did not seem to recognise or understand true/actual infinity. If he did, he would not say there is no set of all cardinalities. Clearly, the set of all cardinalities is Existence/Actual infinity. Again, actual infinity makes potential ‘infinity’ possible. How clearer can it get in terms of what contains the set of all cardinalities?”

What contains infinity? Infinity (and there are no different sizes of infinities)
What contains semi-infinities? Infinity (and there are different sizes of semi-infinities)"

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The analogy with Being and Nothingness is profound here, as existence becomes pre-m-ordinal.

However, that analogy begs the question, for it is based on the assumption that the law of contradiction is phenominal, therefore reducible.

The phenomenal reducibility of contradictory partial infinities, can nor induce the idea of a totality of actual infinities. On the contrary, an infinite assumption of a self inclusive set, needs to be asserted, in order to come up with infinite sets of actual infinities.

Heidegger and Foucault take difference between pre-mordial and pre-mordinal interpretations.

Must confess, this significance is new to me, and would need to reserve judgement upon gathering more enlighted search. Even the question of the reason for significance arising from more archaic sources is kind of anathema.

Even in thought and notion an apple cannot be infinite, I will try to demonstrate this.

There has to be one existing thing that contains all existing things within it. Only actually infinity can non-absurdly fit this description, not our universe. Our universe had a beginning, that which is actually infinite, has no beginning and no end. If we say that our universe had a beginning, in what did it have a beginning in? Non-existence?

If we describe that which encompasses all existing things (including itself) as being Existence or Infinity (capitalised E and I to signify that I am referring to the one true Infinity and Existence), then we do not run into the paradox of something coming from nothing.

If Existence/Infinity was temporally finite, then that it implies it came from non-existence. If Existence was spatially finite, then that implies it is surrounded by non-existence (which implies the existence of non-existence and that is absurd). If there is an end to Existence both internally or externally, then that implies Existence does not encompass all existing things because then It could not be described as encompassing Itself.

Existence/Infinity is the only thing that is Omnipresent. Everything else is semi-omnipresent (semi-infinite or finite) at best. You cannot conceive of an infinite apple because that would be like conceiving of an omnipresent apple. That would be like equating Existence/Infinity with apple. At best you can conceive of an apple that is semi-infinite (or potentially infinite as Aristotle would call it). Just an endless expanse of appleness. This semi-infinite apple, is sustained or contain within that which is Infinite/Existence/Omnipresent. The apple itself, is not Infinite/Existence/Omnipresent. An infinite apple amounts to the paradox of one thing being two different things at the same time. In this case, it amounts to the Infinite being semi-infinite at the same time. The Infinite cannot be anything other than Infinite and Existence cannot be anything other than Existence. Everything within It can change. Finites can become semi-infinite, semi-infinites can become finite etc. But Existence or the Infinite Itself, cannot change.

Which of us is confused on this issue?

Are you talking about an item being infinite in size or an infinity of them existing?

Different degrees of large are still all large. Different degrees of murder are still murder.

I think I understand where you’re coming from. Infinity has clear semantical value. Whether we choose to call this semantical value a number or not, is another matter. I know it encompasses all numbers and all things. Thus, I do think it to have a measure. That measure being Infinite.

These other “infinites” are just semi-infinites in my opinion and you can have different semi-infinites with some of them containing more items than others. You can have more than one semi-infinite or semi-omnipresent thing. But can you have more than one Infinite or Omnipresent thing?If no, then why label that which is endless but has a beginning as infinite?

I completely agree with you here. Which is why I do not like the label potential infinity. Potential omnipresence is absurd, just as potential infinity is absurd. Something is necessarily Omnipresent and Infinite (Existence). Nothing can become Omnipresent or Infinite from a non-omnipresent/infinite state. Hence why I use the labels semi-omnipresent and semi-infinite to pick out those semantics. Semi-infinite and semi-omnipresent are not and never will be Omnipresent and Infinite.

The book has changed into something else. It has not exited Existence. Is it not absurd for something to exit Existence? From where will it exit to? To where will it exit to? Non-existence?

There is no nothing or non-existence. Married bachelors, round squares, nothingness/non-existence, are all absurdities. They are all absurdities because they can never exist and have never existed. That which is absurd/paradoxical/false/hypothetically impossible, is not true of Existence.

Yes, but only in relation to us. Infinite is still Infinite. We describe convergence towards infinity in one direction in relation to us as endlessly becoming smaller and smaller (though never actually becoming Infinitesimal). We know that an endless swimming pool can exist because the pool converges towards Infinity in multiple directions, yet it never reaches Infinity. Yet it is still not finite. Hence why I think it should be called semi-infinite. I can be immortal and in this swimming pool and forever swim forwards within it, and it will never come to an end purely because Infinity sustains me and it. So the pool is endless because Infinity is truly endless and accommodates it. The pool and I had a beginning, Existence/Infinity did not.

For something to be a state or a situation, it has to exist. If this proof does not satisfy, then consider reading the first two posts in the following link:

philosophyneedsgod.wordpress.com

Having no upper limit or lower limit is a semantical component of True/Actual Infinity. Once this semantic is acknowledged, a set of all sets, or a set of all cardinalities, can be acknowledged. Where this semantic is rejected, then Existence is paradoxical on a foundational level. As in we do not have a set of all sets. It is absurd for there to not be a set of all sets.

You cannot have an infinite number of people standing in front of you. You can have an endless number of people stand in front of you, but that which is endless is not necessarily the same as that which is infinite. You can forever move forwards and move past these people that are standing in front of you, but you can never reach infinity. Thus, the number of these people that are in front of you, is not infinite, nor does it amount to or reach infinity.

They mean the exact same thing.

There is no “infinity” to reach. Infinite means endless. There is no end to the endless. If you have an endless line of people in front of you, you have an infinite line of people in front of you. In either case, you cannot ever reach the “end” or “infinity” because there is no end and there is no infinity to be reached.

Fading from existence does not require that anything go out somewhere else.

Are you saying that the Roman empire still exists? Living dinosaurs?

If something disintegrates and the remains become something else, what they were no longer exists instead what they are exists - temporarily. Every physical thing is always changing into what it wasn’t and from what it was. Everything is ALWAYS leaving existence as new things form from their remains.

Are the Roman soldiers still walking around? They were. Are they still Roman soldiers? They cannot be. And that page from that book is no longer a page in a book. They all became only the remains of their former existence.

Ok, we agree that one aspect of Infinity is endlessness. I think it meaningful to say that you can have something that has a beginning but no end. I don’t want to call this infinite because it has a beginning. So I call this semi-infinite and I reserve the label infinity for that which has no beginning and no end. Which can only be Existence or that which is Omnipresent. The Omnipresent can accommodate an endless line of people, but it cannot accommodate another Omnipresent/Truly Infinite being.

Also I am in agreement with you now in that there is no set of all sets that are members of themselves. For something to be classed as a set, it must contain at least more than one thing. There is only one that is a member of itself. Thus there cannot be a set of all sets that are members of themselves. That’s like saying there can be a set of Omnipresent beings. My apologies for not realising the soundness of your point earlier.

The past does not go out of Existence. It always exists in Existence. It does not exist in our reality in our present time, but it certainly exists in Existence. Thus our past still exists in Existence. We don’t exist in our past, we exist in our present and move towards the future. We cannot go to the past as the past cannot occur in the future. Which essentially means time travel is absurd. But the past not being in Existence, is also absurd.

I strongly recommend a read of the first two posts here:

philosophyneedsgod.wordpress.com

You are imagining things I didn’t say, and then trying to correct them… please don’t!

But therein lies the problem, that we are dealing with theories not facts, and yes… I can imagine infinite apples, but infinite apples can never be ‘a thing’ even though I have imagined them to be so. Are you trying to convince me otherwise? because that is all that can happen here, in being convinced that another’s mind is telling me that what I have imagined in my mind, is wrong.

We do not know if the Universe is the container of all things, or if the Universe itself is being contained, so again… we can but only theorise, with the information and knowledge that we have. But then, the question of where did the Universe’s container originate, arises?

Edited to add: I have now thought of the many different ways that the above could be and work. I will wait to see what you have to say on all the above. Thanks.