## Do you believe there can be an Actual Infinite

• 1
the condition for an infinite to exist in the first place is that it expands without an end, but the mere fact that we can conprehand the idea, the conception of something called infitiy serves as a proof that something like that does indeed exist if not materialy then spiritualy in our minds maa it be small or big.(beacuse infinity doesnt have to be in direct correlation with size).
• 1.9k
I was thinking of trying to get an article published on Infinity. I thought I'd post it here first. Any thoughts?

Infinite Confusion

Infinity has been a source of fascination and confusion for 1000’s of years. Here is a brief review of the history of infinity and try (hopefully) to clear up some of the confusion it causes in maths and the sciences.

Some History

The earliest reference we have to infinity is from the Greek philosopher Anaximander who used the word ‘Apeiron’ to refer to limitless, unbounded or indefinite.

The great Greek philosopher Aristotle subsequently made an important distinction between two kinds of infinity. ‘Potential Infinity’ he described as a iterative process that can potentially be carried out for ever never actually is. Examples are counting or walking. ‘Actual Infinity’ is then defined as the results of carrying out the iterative process for ever. Aristotle felt that Potential Infinities were OK but Actual Infinities were not allowable.

Still with the Greeks, Zeno of Elea (born c. 490 BCE) is famous for his paradoxes of motion. An example paradox is the story of a foot race between Achilles and a tortoise. The tortoise asks for a 50 meter head start and is confident of victory; in order for Achilles to catch the tortoise, he first covers the 50 meters. By that time, the tortoise has moved ahead another 5 meters. By the time Achilles has moved another 5 meters, the tortoise has moved ahead again and so on; with the conclusion that Achilles will never catch the tortoise because he must perform an Actually Infinite number of steps to do so.

The simplest solution to Zeno’s paradoxes is to assume space comes in discrete fixed-size chunks (rather than continuous space) so that Achilles then only has to cover a finite number of steps to catch the tortoise.

Whilst it is not mentioned in the Bible, christian theologians have traditionally attributed infinity to God, stressing the unbounded nature of God’s power. To deny God anything was seen as belittling God.

Georg Cantor, the german mathematician responsible for much of modern set theory, was a devout Lutheran and believed his work on infinite sets was communicated to him directly by God.

Infinity in maths

Infinity is by its very nature unbounded and therefore not well defined. Infinity lacks a start or end; what other object lacks starts and ends? These ill-defined and unnatural characteristics of infinity make it prone to causing paradoxes (as we’ve seen with Zeno).

First thing to note that infinity is not any sort of normal number or quantity:

There is no quantity X such that X > all other quantities because X+1 > X.

To reinforce that infinity is not a quantity, it also behaves unlike any normal quantity under the operation of the basic mathematical operators. Adding, subtracting, multiplying and dividing infinity all yield infinity as the result:

1 + oo = oo implies:
1 = 0

In Calculus, the limit concept is used to describe an expression approaching, but never actually achieving the value. The limit concept is used with infinity for example, it is common to write:

lim 1/n = oo
n->0

Its important to realise that the limit never actually evaluates to infinity; it is always a finite number that approaches but never reaches infinity. So strictly speaking, its more accurate to write:

lim 1/n ~ oo
n->0

Geometrically, infinity is a source of confusion. How many points can you get on a line segment of length 1? The traditional answer is an actually infinite number of points. This does not hold up too well under closer examination. A mathematical point is defined to have length zero. So the number of points in the interval is:

(segment length) / (point length)
= 1 / 0
= Undefined

Something with length 0 can’t exist so it seems the mathematical definition of a point is contradictory. Using a redefinition of ‘point’ to have a non-zero length, we can see there are always a finite number of points in a segment. As the point size decreases; the number of points tends to but never actually reaches Actual Infinity. So the number of points on a line segment is an example of Potential Infinity.

Also geometrically, actual infinity is not constructible. For example, it is impossible to construct a line segment with the property that it is longer than all other line segments.

In set theory, the actually infinite is defined to exist by way of the ‘Axiom of Infinity’. So set theory does not prove actual infinity exists it merely assumes that it does. An axiom is meant to be a self-evident truth. Its questionable whether the existence of a set with an actually infinite number of members is a self evident truth. It has to be remembered here that set theory was devised in the late 1800 in a still heavily religious society. Cantor and company regarded it as self evident that God was infinite and required mathematics to reflect this.

Infinity In Science

Science is a two-pronged subject; the theoretical and empirical. As has been mentioned, theoretically, actual infinity is on somewhat shaky ground. Traditionally, science treats actual infinity as indicative of a error in calculation. For example the infinity/singularity at the heart of the Big Bang is regarded as indicative that the theory of Relativity has broken down.

Empirically, things look no better. There are no examples of actual infinity in the material world that we know of.

There are some unknowns such as how far space goes on or how far time goes back; but these are not evidence for the Actually Infinite, merely just a lack of evidence either way.

Continuous space and/or time are sometimes used as an example of the actually infinite, but modern science is trending in the direction of the discrete. Matter is discrete. The Bekenstein bound (https://en.wikipedia.org/wiki/Bekenstein_bound) expresses a limit on the information content of a region of space and strongly suggests discreteness of space.

Another way of thinking about it is to consider a 1cm cubed volume of ‘continuous’ space; it will be graduated with infinite precision as that is the definition of continuous; the positions of particles within it will be know with infinite precision; which equates to infinite information in a finite volume. Also then consider a 1 light year cubed volume of ‘continuous’ space; it will also be graduated with an identical infinite precision (as that is the definition of continuous). This suggests the information content of the two volumes are both infinite and similar. Seems nonsensical; hence discrete space. A simpler argument applies for discrete time.

There is some uncertainty in science over whether the universe is finite or infinite in time and space. There is a simple argument against an infinite universe; if it does not have a start, it can’t exist. So this argument implies the universe has a start in time (and time itself has a start).

The word ‘Eternal’ is often uses with infinite time and has two meanings:

Eternal Outside Time - existing for ever outside of time
Eternal In Time - existing for ever within time

The first meaning of eternal does not require actual infinity and is compatible with Einstein’s 4D space-time view of the world. The second meaning does require actual infinity and leads to paradoxes, for example:

- Say you meet an Eternal (in time) being in your Eternal (in time) universe
- You notice he is counting
- You ask and he says ‘I’ve always been counting’
- What number is he on?

The problem here is ‘Eternal In Time’ - it has no start so it is undefined/cannot exist; hence the paradox.

The measure problem from Cosmology is another paradox due to an infinite universe:

- Assume time extends back for ever.
- If it can happen it has happened.
- An infinite number of times.
- No matter how unlikely it was in the first place!
- So all things have happened an infinite number of times.
- So all things are equally likely.

Another argument against an ‘Eternal In Time’ universe is the 2nd law of thermodynamics: If the universe has been around for ever then it should be in thermodynamic equilibrium by now. But the universe is not in thermodynamic equilibrium.

Closing Remarks

In an article this length, I cannot begin to iterate all the paradoxes of infinity…

Hilbert’s amazing hotel that is completely full with infinite guests; an infinity of new guess arrive and by magic they are all accommodated (https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel)

Cantor's Paradox: ‘The set of all sets is its own power set. Therefore, the cardinal number of the set of all sets must be bigger than itself.’ The set of all sets is an ACTUAL INFINITY so not a completely described set. You cannot soundly reason with it. Leads to the paradox.

Posit an universe infinite in time but finite in space plus some historians. Then there is not enough room in the universe to write down the whole history of the universe!

A paradox is caused by an error in the underlying reasoning; the assumption that Actual Infinity is possible is the cause of these paradoxes in every case.

One finial thought; how exactly is Actual Infinity and the materialistic world view comparable?

Finite regards…
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Aristotle felt that Potential Infinities were OK but Actual Infinities were not allowable.

If you allow that the eternal is infinite, then Aristotle does allow for an actual infinite, because he demonstrates that anything eternal is necessarily actual. This would appear as an inconsistency in Aristotle's principles.

Whilst it is not mentioned in the Bible, christian theologians have traditionally attributed infinity to God, stressing the unbounded nature of God’s power. To deny God anything was seen as belittling God.

God is said to be eternal and actual, so it is the associating "eternal" with "infinite" which renders an actual infinite in the form of God. If we switch "infinite" for "eternal", then instead of being actual and eternal, God is actual and infinite.

The word ‘Eternal’ is often uses with infinite time and has two meanings:

Eternal Outside Time - existing for ever outside of time
Eternal In Time - existing for ever within time

"Outside Time", being the sense of "eternal" which is generally associated with God, does not equate with "infinite". What it implies is that time is bounded (there is something outside time) and therefore time is not infinite. So "eternal" in this sense does not imply "infinite", it implies that time is finite, and there is consistency in Aristotle's principles which state that the eternal is actual, and that the infinite is potential.
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his would appear as an inconsistency in Aristotle's principles

Good point, time is so mysterious and ethereal; wonder if he somehow thought an infinity of time was different to infinity of other kinds. I see all instances of Actual Infinity as fictitious; time and space still have to obey common sense.

God is said to be eternal

So it says in the Bible to; but it does not say which kind of eternity and theologians have differed on the answer. It looks like timeless eternal is the answer; that just leaves the question of what timeless existence/beings could be like. Photons are timeless yet they change position and wavelength. They move at the speed of light; maybe everything in the timeless realm moves at the speed of light.

"Outside Time", being the sense of "eternal" which is generally associated with God, does not equate with "infinite"

Agreed. God (if there is one) is finite. I don't think that takes anything away from God; infinite is simply not possible and even an omnipotent God cannot perform the impossible.
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The infinite sets the premise for continuity in truth; no continuity, no truth.

Finiteness and infinity do not contradict.

Take the line for example.

***Ideally this would be best observed with pictoral examples, but words will suffice...I hope...

1. It observes a finite degree of one, where the line as infinitely projecting observes 1 as continuous.

2. The line as projective, cannot project unless there is somewhere for it to project too or from, hence the line must invert if it is to project anywhere. This inversion so to speaking observes the line going in an opposite direction from the beginning point. This is considering the point the line projects towards is the same point it is projecting from as all 0d points are the same.

3. This leads to two infinities considering the line is infinite. So now the line as multiple relative lines, through the inversion of 0d point space, is now finite yet stemming from the same infinite source. Finiteness is multiple infinities in these respects and now we have a premise for infinity to be quantified.

4. These two lines are simultaneously half of the original line. So with the inversion of the line, which for definition's sake I will call "folding", comes simultaneously quantitative expansion (multiplication) and contraction (division as fractals). This line as two lines, still however is one line considering the line is an infinite numbers of points between polar points. Hence, we can observe 3 lines in one.

5. These two lines, stemming from 3 points (one at each polarity and one in center), must continually project past there origins if they are to maintain themselves. However both lines are projecting from an origin of the center point, hence what we observe through the line is strictly the point as void existing through direction...This is considering a line is a point directed towards a point. So with that in mind the point must project towards itself, through the line.

In these respects the points at both ends are directed towards each other as these points are directed away from the center point, while towards each other as extensions of the center point. This is considering point exists through point. These end points directed towards each other effectively multiply the line, considering the line exists as one direction where quantity is specifically directional.

This forms the first angle, with this angle "condensing", as each end point is drawn towards the other. Infinite directions, are observed, or under the terms of the degree: 90 degrees (directions) are observed as 90 directions.

6. The angle as it condenses again form itself into a line, with the angle being 1 direction in itself. Under these respects the line exists as an infinitely condensing angle. In simpler terms the line exists as an angle and the angle exists as a line where one cannot exist without the other. The angle and line are duals so to speak.

7. The angles as a line must go in an opposite direction, considering the premise of the line existing is one off continually inverting directions. So the angle expands, forms into the original triadic line and contracts in the opposite direction.

8. Considering this occurs as a rate of infinity we can observe 4 directions at once (as 6 lines, vertices and horizon are 3 in 1.), through infinite grades of direction or 360 degrees, results in right/left and up/down. So now the line exists as a four directional structure as two dimensions.

9. This structure in turn must project away from its origins where the end points are directed towards each other and the center projects. This results in depth as the object goes forwards and backwards. From one angle this would appear as all lines contracting to a center point than expanding causing the object to seemingly disappear into a point and then expand from a point (or coming and going out of nothingness as it alternates between the dimensions of forwards and backwords).

This folding into "forwards" results in 4 lines as 5 with these four lines being a quarter of the original, hence the "folding function" observes 1/4 and 4 as simultaneous. This happen dually with backwords as well. So at this step we have 3 lines of horizon, vertice and depth where the line is reduced to one sixth.

The problem occurs, considering the line is the only standard of size, the lines are only relatively smaller compared to there prior localization and hence effectively are not shrinking or expanding on there own terms considering each line is infinite.

10. So with depth included we have 6 lines through 3 as 9. Now this is occurring at a rate of infinity with each "geometric object" quite literally being its own time zone. This process occurs further, and would need actual graphs at this point, until you reach infinity as a sphere with the sphere as infinite directions being conducive to 1 in itself.

11. The sphere, relativistivally, becomes a point, considering it must project in one direction and the cycle continues. Hence the point is relatively a sphere and the line is composed of infinite spheres as infinite ones and zeros.

12. Going from the points it may be observed that we have a number progression so to speak, where all numbers are actually infinities. All these steps are mere localization of infintity so too speaking; hence simultaneously proves through a folding process that there are possibility infinite ways to result in a number line.

I will stop here....
• 267

In regards to the contradiction, you observe, of the 0d point having no length and yet the line is composed of infinite 0d points, I believe the contradiction is solvable.

The 0d point is void. Void cannot be observed except through "being", ie the 1d line in this case. Considering the premise of being, stems from a unity of 1 as everything (which does not contradict the atomist perspective considering only atoms exist) what we understand of void is an observation of opposition through separation.

Being and void represent the first dualism, or opposition as contradiction so to speak. Being exists as being. However void cannot be observed in itself, except through a multiplicity of being, where void effectively is reduced to nothing and not a thing in itself. This is considering multiplicity of being is still just 1 being. Void, or the 0d point, as nothing is strictly an inversion of unity of being through multiplicity.

The question of being leads to the question of composition, what exists universal through all being as being itself. Considering all being exists through structure, this foundation is limit, with the line being one of these limits (I will stop this point here to avoid lengthening the argument further.).

Void effectively cancels itself out through its own nature, as the inversion of inversion in one respect leaving the line and circle. In a separate respect this inversion of inversion maintains itself as a constant with this constant nature not only resulting in the line and circle being composed of multiple lines and circles, but fundamentally resulting in continuity.

This continuity sets the premise for direction where direction and unity are inseparable, considering the void of void results only in being with this being existing as 1. In these respects the line and circle become both observations of 1. The line sets the premise for 1 direction as a part of all directions, with these all directions existing as the circle and is conducive to 1.

Considering these infinite 1s exists as one with each other of these 1s being composed of infinite ones, what we observe is both one and many infinities with this dualism being solved under a triadic element of "limit" (considering geometry sets the foundation for quantity).

Finiteness is multiple infinities, with infinity being one. This cannot be proven or dispproven without using these same variables as part of the framework of proof, hence all proof lies in these limits.
• 17
Infinity is a process of something that takes forever. The word or an idea what you're looking for is limitlessness. Limitlessness can only be if there's no beginning and no end. The ultimate word is limitlessness and not infinity.
• 267

Good point, but we are stuck with a paradox, considering the limits of language (which give rise to the point you are observing) are progressive in nature. One definition progress to another, and so on, with this continual progression leading to a further paradox of cycling.

All of this reflects the premise of the thread relative to "actual infinities".

Using the dictionary as an example: I look up x definition and it leads me to y. I look up y and in simultaneously leads me back to x while separately progressing to z. X is connected to z through y. And the process continues, through language, as potentially infinite considering the definition of language itself (through the dictionary) inevitably leads back to limit and limitless; necessitating that language is an extension of everything as everything can be summated under limit and limitlessness.

Now assuming you agree with the above, what we understand of "limit", "limitless" and "infinite" are funda,mentally connected.

What we understand of "limitless" is an absence of limit necessitating that limit exists as a negative exists relative to a positive. We can see this with potentiality fundamentally being a formless actuality, or absence of actuality.

All limit that exists must be continual, with this contuity fundamentally being an absence of change. The problem occurs in the respect that the limit effectively must change if it is to continue. We see this in the evolution or organisms and in their basic reproduction cycles. The problem is that this continual change, as absence of limit, is in itself a limit considering continual change is...well "continual". Hence the continuous that necessitates limit, is the continuous change of a limit as a limit in itself.

In these respects, limit exists through limitless with this limitless in one respect negating itself into a limit. In a separate respect this continual negation, resulting in limit, effectively results in the limit as limitlessness.

In these respects the limit exists through multiple limits as one (we can see this in the continuity of a species at a larger level, as multiple parts proprogating as one).

Looking at the nature "infinite" we can observe a negation of finiteness. This follows the same format of the limit. However what we understand of finiteness effectively is a part or locality which exists relative to another part or locality. This necessitates a multiplicity where a part as composed of and composing further parts is an "actuality" so to speak where the part is a continuity of the parts it is both composed of and composes. This actuality as locality exists as a limit in itself.

Now the actuality cannot move, hence exist, unless there is a potential localization or absence through which it can move. This may sound abstract so here is an example. An athlete cannot progress in his fitness unless he has the potential to increase. However this potential nature, is in fact a change from his actual nature in one respect while being formless in another. The athlete can only increase in fitness if he has the ability to change from one state of being fit to another. Infinite potential growth is infinite actual growth as change.

Under these terms all potentiality is an absence of form which negates itself into actual form. So continual finiteness, or parts/localities, is a unification of finiteness as continual. The continuity of finiteness is in itself infinite so to speak, where finiteness as infinite and multiple observes itself as multiple infinities through 1 infinity.

From here we see a continual circularity between finite and infinite which effectively acts as either a literal or abstract limit through the "circle". This circle as a limit to this alternation of "concepts", if not realities, must in itself become continual hence we cycle back to limitlessness.

Under these terms, and this "argument", limitlessness and limit are effectively one and the same where any difference as seperation merely being an approximation of unity.

Language, the source of the problem you are observing, is determined by a form of limitless multiplicity which any universal language, that continually, having to be built on symbols or words which are few in number (with the increase in word definitions comes an increase in complexity and confusion) and represent a large number of limits under them by effectively unifying them.

The problem of limitlessness, is a paradoxical limitless number of definitions for it.

.....hopefully the above makes sense.
• 1.9k
Infinity is a process of something that takes forever. The word or an idea what you're looking for is limitlessness. Limitlessness can only be if there's no beginning and no end. The ultimate word is limitlessness and not infinity.

If something has no beginning or end, it does not exist.

limitlessness = unbounded = undefined = does not exist

Actual infinity does not exist.
• 267
Hence the statement will eventually become untrue as it is finite.
• 11
If space is finite, then what is beyond the end of space? I think it'd be more space. But I also accept nobody knows if the universe is infinite or not. It might even be something else.
• 1.9k
If space is finite, then what is beyond the end of space? I think it'd be more space. But I also accept nobody knows if the universe is infinite or not. It might even be something else.

There is nothing beyond; no space, no time, just nothing...

Logically the universe must be finite:

- Its expanding from a point (the Big Bang) so it must have a finite radius
- Anything expanding must be finite else it would have nowhere to expand to.
- Actual infinity is impossible (see the rest of this thread)
• 267

Beginning and end are just points of relation stemming from a center point with all beginnings and end as center points in themselves. Entropy proves the end of one physical phenomena is the beginning of another, and beginning and end exist simultaneously.

No beginning or no end only gives premise to a constant center. An example would be a sphere, where any point on the circumference is a center point.

Limitlessness is continuity with continuity being the foundation of all limits. Using 1 as an example it may be observed that as a finite entity it must continually project through all other numbers if it is to exist as a constant. This continual projection of 1 ad-infinitum neccessitates infinity exists as 1.

Finiteness is merely multiple infinities.
• 267

- Nothingness cannot be observed on its own terms except relative to another form of being, hence nothingness is not a thing in itself but rather a statement of relation. Because nothingness can only be observed through relation, and not on its own terms, there is no nothing but rather grades of "being" as Multiplicity of "being".

- Considering the premise the universe expands and contracts through a point, necessitates the universe acting through a repetitive frequency where it cannot be observed as being the only universe considering all expansion and contraction exists as part of nature and physics as a continuum.

- Actual infinity is possible through linear time, with the end of one time being the beginning of another, while being composed of infinite time zones in itself consider time is merely the relation of parts. All lines being composed of infinite lines is an example of this along with irrational numbers continually repeating a sequence.

- Finiteness is not logical on its own terms as no truth statement can be made without it inevitabling canceling itself. To say "there is no infinity" would necessitate the statement canceling itself out eventually leaving it as false and the existence of infinity as true or the statement would cancel itself out an be replaced with another statement ad-infinitum making a perpetual negative that in itself is infinite relative to a positive infinity.

- Where are you planning to publish exactly, out of curiosity?
• 1.9k
Actual infinity is possible through linear time

If you take away the start of time (like Monday) then the rest of time does not exist (like the rest of the week). So time has a start.

A time interval is not composed of a infinite number of moments; a moment has length 0; so (interval length)/0 = UNDEFINED

I believe time has an end too; it exists in 4d space time as a finite object so it must have an end. The end could be coincidental with the start.

I have no idea where to publish, any recommendations?
• 267
Actually time has points of inversion, which may be equated relatively to beginning and end points, but fundamentally are both as points of center.

This concept may appear confusing so I will elaborate.

What we under of Monday is only a beginning of a week when setting it as a premise for a work or religious cycle of time. The Monday as the beginning of one week is the end of a week for another. For example tomorrow's Monday is both the beginning of a week and end of another considering one beginning is quite literally the end for something else.

Monday exists as both beginning and end with beginning and end existing as opposing poles through a mediation of movements along a timeline (in this case the week). Monday is simultaneously both beginning and end with beginning and end being directional statements of relation. Under these terms Monday, is merely a point of inversion where a unified timeline (a week) is inverted into multiple timelines (weeks).

In these respects Monday is neither beginning or end, but rather a center point of change that divides timelines in repititve ratios where the reoccurring of every week occurs much like a repetitive frequency.

As a point of inversion, and using the linear nature of time as an example, the line as composed of infinite points is effectively composed of infinite lines.

A moment as multiple timelines within it and cannot be reduced strictly to zero, considering all moments in time are compose of relative movements. For example, I may remember s moment in time as relatively "still", but this "stillness" only takes into account a perceived absence of movement relative to another.

I may remember my hand being still because my hand was the fastest movement observed, an all movements relative to my hand slowed down. I may also remember a bee being still in the air because the rate of movement in its wings was so fast as to not be observed.

Everything may also be still in the room, but the light waves are so fast we assume no movement.

Hence with an increase in movement comes an increase in "stillness" as the moment, however that which sets the standard of speed forms its own time zone so to speak, where that which moves slower is a separate time zone.

A particle may have 3000 revolutions per second. B particle may have 1000 times that. A and B are defined by the movements of there parts. A and B are there own time zones considering the frequency of revolutions define them. However B may be used as the standard considering it having the highest rate of movement allows all other particles rates to fit inside its own rate. Hence particle B as a standard of time observes a potential unity of relations from which everything is judged.

In simpler terms a moment is composed of further moments and cannot be equated to point 0 as a moment is a timeline in itself.

1 divided by zero results in undefined as infinity.
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