Can There Be An Infinite Number Of Beings? I Don't Think Its Possible!

[MindOverMatter2]

A point just refers to a specific location in space. Not only are there an uncountable infinity of points in space, there are infinitely many points (also uncountable) on a line segment. This means that for any two locations that are not the same location, there are infinitely many points in between the two locations.

This has the illusion of a very good counter arguement; and it is a good point. Pehaps, when dealing with space, you might be able to show an potential infinite number of points, as in to say that you keep on dividing space given the idea of space and still end up with space. Space has no parts in the same sense that a pie can have parts and can be broken in to smaller parts. But there is evidently an ontological distance that can be quantified. Your arguement has no bearing on whether space can be actually infinite in the objective sense that we are using the term. What i mean to say here is that, while there may very well be an infinite number of “points” between two points abstractly speaking, ontologically speaking i can travel a “finite” distance between two points. Metaphysically speaking there is a finite distance between two points, otherwise i could never reach any point. I would be stuck in one point. Thus there is obviously not an infinite number of distance between two points. The same is true of the actualisation of potential beings; you cannot actualise an actually infinite number of beings.

A metaphysically infinite number of beings is not the same thing as the infinte that you describe.

 

[MindOverMatter2]

Metaphysically speaking there is a finite distance between two objective points, otherwise i could never reach any point. It would be impossible. I would be stuck in one point; never being able to proceed to another point because it trully is an infinite distance away. Also a spatial point that has zero distance, cannot objectively exist, since there is no distance and thus no points divided by distance.

An “infinite point” is not physical.

 

[Leela]

An “infinite point” is not physical.]

A point, mathematically, is without dimension. It has no length or width. Points are not considered physical objects. I am willing to accept that there are not be infinitely many physical objects because physical objects are presumed to have some measureable size as is the uiverse itself, so there can only be a finite number of objects of a given size within a finite space. This seems obvious, and I don’t understand the importance of saying it is true.

Let me add that this definition of a physical being as having some measurable size rules out a lot of reality–even physical reality such as energy and light and subatomic wave-particles. Once we get down to a subatomic level, the notion of comparing an object to a ruler to measure it starts to become incoherent. So I wonder if this question is even meaningful for whatever purposes the poser of the question had in mind.

Best,
Leela

 

[chrysostim83]

I don’t see how that could be possible, but I’m sure I know an even smaller amount of set theory. Could you explain?

The set theory stuff will look a bit strange, but I’ll do my best to explain what I had in mind. When Georg Cantor first started looking into transfinite numbers, he noticed there are actually two different ways of defining them. The first way, which yields ordinal numbers, answers to our intuition of counting by arranging a set into a long list of items: ordinal numbers simply measure the length of such lists. For example: (1) if a set can be arranged in a list that has three items, then the length of that list is the ordinal number 3; (2) the length of list with exactly one item for each natural number (0, 1, 2, etc) is, we’ll say, w; (3) the length of list with exactly one item for each natural number plus one additional item tagged on at the end is w + 1. This shows the first way an infinite being could count an infinite set. For suppose the list of years of the beings life has length w + 1 and that the list of items the being has to count has length w. Then the being will have counted everything by the time year w + 1 arrives.

The second way of defining transfinite numbers, which yields cardinal numbers, answers to our intuition of counting by finding a correspondence between two sets. For example: (1) the word “for” has three letters because of the correspondence given by f <-> 1, o <-> 2, r <-> 3; (2) the set of squares of natural numbers has the same cardinality as the set of natural numbers because of the correspondence given by n^2 <-> n; (3) the set of real numbers has a greater cardinality than the set of natural numbers. This last example is really what got set theory going, because for the first time it became clear that infinity isn’t the generic, well-nigh pathological concept it had always appeared to be: it actually contains a highly structured universe of ever larger infinities. Two elements of this universe of infinities are |w|, the cardinality of the ordinal number w, and |w|+, the next larger cardinality. Now, it turns out (it would take quite a lot of space to explain why) that if an infinite being lived |w|+ many years, it could not fail to finish counting a set of cardinality |w|.

Those are, at least, the two ways I had in mind when I wrote that an infinitely long-lived creature might count an infinite set. Just in case you’re interested in further exploring this subject, Hrbacek and Jech’s introduction covers the material you’d need to judge these proposals.

We’d have to then ask how it could be proven that it is impossible for us to ever know (with an infinite amount of time), the physical universe. The assumption is that a study of physical laws and materials is not obscured or impenetrable to human reason.

But this is also how we define “objective reality” or “actual things” or “physical universe”. If something cannot ever possibly be verified or actualized through physical means, can we say that it is part of objective-physical-material reality?

There are some pretty complicated issues here. To the first part I would say that science isn’t really on the look out for just any old fact. Physicists, for example, try to work out the physical laws of space-time, matter, energy, and so forth, and it could turn out that these fundamental laws are knowable—and thus, reality would still be intelligible—while certain comparatively irrelevant facts (such as the cardinality of the universe) are not.

The second part brings up difficult questions (e.g., what counts as verification?, who gets to do the verifying?, if all verification is only probable, then doesn’t physical existence become vague at the edges?, etc). Fundamentally, however, the issue here is the age-old rivalry between metaphysical realism and anti-realism. Unfortunately, all I have are realist intuitions. Perhaps we should conclude something like this: if you’re a realist in metaphysics and mathematics, then you’ll think an infinite set of physical objects is possible, but if you’re not either of those things, then you won’t. William Lane Craig’s anti-realism comes into focus here.

I can’t resist adding, pace Leela, that a finite universe could contain an infinite number of extended objects: simply choose the objects so that the n-th object takes up no more space than 1/2^n cubic meters.

 

[Leela]

I can’t resist adding, pace Leela, that a finite universe could contain an infinite number of extended objects: simply choose the objects so that the n-th object takes up no more space than 1/2^n cubic meters.

I stand corrected and blushing.

 

[reggieM]

Thanks for this detailed reply, chrysostim. I definitely do not understand it so please be patient with my question.

(2) the length of list with exactly one item for each natural number (0, 1, 2, etc) is, we’ll say, w;

As I see it, this attempts to separate “length of” with “count of”. It seems like a sleight-of-hand trick. The idea that one could measure from a beginning point to an indeterminate end point is irrational and impossible, as I see it. In order to come up with the value of w, one has to choose an end-point in the length of the list, right?

(3) the length of list with exactly one item for each natural number

As above, how does one measure the length of something that has no end-point?

For suppose the list of years of the beings life has length w + 1 and that the list of items the being has to count has length w. Then the being will have counted everything by the time year w + 1 arrives.

The list of years of an immortal being’s life has the length of ??? Perhaps we’re saying something different. I accept that chrysostim’s soul will exist forever, therefore his soul has an infinite quality. Now, I measure how old he is, starting from his birth. Thus, I have counted an infinite string? No - I have counted a finite string. To count the infinite I need to count how many years chrysostim will live in the future, right?

Now, it turns out (it would take quite a lot of space to explain why) that if an infinite being lived |w|+ many years, it could not fail to finish counting a set of cardinality |w|.

I have to assume that these arguments have considered the most basic, logical objections that I can think of – and since it’s too complex to post the reasoning here, I will have to just bow out of this and try to understand the proposal. Thanks for the reference.

I can’t resist adding, pace Leela, that a finite universe could contain an infinite number of extended objects: simply choose the objects so that the n-th object takes up no more space than 1/2^n cubic meters.

I think this is similar to MOM’s comment earlier today regarding the claim that a line, for example, contains an infinite number of points. That is false because one would never be able to identify any one point on the line (since the point has no determined size) and it would take forever to reach the second point (since there is an infinite distance between points).

I think about Catholic sacramental theology regarding the Eucharist. We know that Christ is truly present in the Bread-made-Body, even in fragments. But, however, fragments can become so small that they are no longer Bread (the proper matter of the sacrament), thus we do not commit sacrilege by steping on microscopic dust of the Host, etc.

I see it the same with a physical object. If it took an infinite (never ending) amount of time to identify if the object exists, then the object does not exist and would be equivalent to any other state of non-existence. Its the same as “nothing”.

A physical object must have some means of being identified as such and being distinguished from nothingness.

 

[chrysostim83]

As I see it, this attempts to separate “length of” with “count of”. It seems like a sleight-of-hand trick. The idea that one could measure from a beginning point to an indeterminate end point is irrational and impossible, as I see it. In order to come up with the value of w, one has to choose an end-point in the length of the list, right?

You’re definitely right here. The problem of measuring a list with length w can’t be solved with finite materials; to solve this difficulty, set theory usually comes equipped with an axiom of infinity which postulates outright the existence of w. Perhaps, then, these set theory ideas won’t work at all because they merely beg the question.

I think this is similar to MOM’s comment earlier today regarding the claim that a line, for example, contains an infinite number of points. That is false because one would never be able to identify any one point on the line (since the point has no determined size) and it would take forever to reach the second point (since there is an infinite distance between points).

Zeno’s paradox must have one solution or another. I would say there are (or might be) an infinite number of points between A and B, but the distance from A to B is a different concept from the number of points contained between A and B, so it is possible for the former to be finite while the latter is infinite.

I think about Catholic sacramental theology regarding the Eucharist. We know that Christ is truly present in the Bread-made-Body, even in fragments. But, however, fragments can become so small that they are no longer Bread (the proper matter of the sacrament), thus we do not commit sacrilege by steping on microscopic dust of the Host, etc.

I see it the same with a physical object. If it took an infinite (never ending) amount of time to identify if the object exists, then the object does not exist and would be equivalent to any other state of non-existence. Its the same as “nothing”.

A physical object must have some means of being identified as such and being distinguished from nothingness.

This is fascinating point. I’m not so sure about the second paragraph there, because it sounds as if “we” would have to be able to distinguish the object from non-being in order for the object to exist. It seems to me to be enough that the object be distinguishable from non-being—exactly what kind of being is capable of making out the distinction and how long it takes shouldn’t matter. Still, the first paragraph is suggestive of an argument like this:

1. 
   *]Every physical object has a substantial form (premise and third paragraph of above quote)
   *]An entity cannot have a substantial form unless it is at least as large as some particular physical object (premise, supported by analogy with host: the smallest physical object will be the least fragment of bread that can still be considered bread)
   *]Therefore, every physical object is at least as large as some particular physical object (by 1 and 2)
   *]The total size of the universe if finite (premise, backed up by science)
   *]Therefore, there are only finitely many physical objects in the universe (by 3 and 4)
   
   Now this is an argument I could accept for the actual finiteness of the universe. It relies on a property of physical objects, viz., their need for a substantial form or to be one sort of thing and not another, and so it doesn’t destroy the vast expanses of modern mathematics. (Craig’s argument does do that, since he rules out all actual infinities, even the conceptual ones.) One especially nice feature is that it depends on the contingent fact that God chose to create a finite universe. It’s as if He wrote his signature in its very finiteness.
   
   The only dubious thing about the above argument is premise 2. Perhaps by becoming too small to be bread, the fragment of bread becomes something else, like a crumb. There seems to be no telling how small a portion of matter needs to be before it ceases be able to receive a form.
   
   PS: Leela: no blushes necessary. An infinite number of objects in a finite space is counterintuitive.

 

[MindOverMatter2]

1. 
   *]Every physical object has a substantial form (premise and third paragraph of above quote)
   *]An entity cannot have a substantial form unless it is at least as large as some particular physical object (premise, supported by analogy with host: the smallest physical object will be the least fragment of bread that can still be considered bread)
   *]Therefore, every physical object is at least as large as some particular physical object (by 1 and 2)
   *]The total size of the universe if finite (premise, backed up by science)
   *]Therefore, there are only finitely many physical objects in the universe (by 3 and 4)
   
   Now this is an argument I could accept for the actual finiteness of the universe.

1. 
   Of course, i am talking about quantifiable measurable beings.
   

   The only dubious thing about the above argument is premise 2. Perhaps by becoming too small to be bread, the fragment of bread becomes something else, like a crumb. There seems to be no telling how small a portion of matter needs to be before it ceases be able to receive a form.

   We can see that things are larger or smaller than other things; thus there must be a smallest thing before a thing becomes nothing. It wouldn’t make any sense that a thing can be a particular size in comparison to another and yet identical in size in so far as there never being a smallest dimension to a thing. In reality, it wouldn’t really have a size, and thus there would be no logical basis for one thing being larger or smaller than another; since the words large and small would be objectively meaningless. Yet we see that things are objectively larger and smaller than other things.
   
   To put it another way. If the circumference of a circle infinitely regresses in terms of smallness, then “smallness” becomes meaningless, because objectively the circle will always have the same size, or rather, the same circumference; since it never reaches a limit. Size becomes infinite. Since the measurement of size becomes meaningless, how is it that the circle has a definite circumference? Size and measurement becomes meaningless and contradictory. It would be the same problem if one said that a circle was infinitely large, because the circle would lose its definition as a circle when that which defines the finite dimensions of a circle is beyond finite measurement. Its definition would objectively disappear. It is possible for a thing to be defined as circle only because of its finite dimensions or circumference.
   
   A measurable form is at its smallest before it becomes zero. In other words there is such a thing as the “ontological one”. For instance, if our universe began to exist, than there has to be a smallest amount of time after point zero, otherwise it makes no sense that we can progress in change after the fact of one if there is already an infinite amount of change between zero and one.
   
   Therefore it is not necessary for us to know or determine the smallest measurable thing, because we know that in order for there to be a definable difference there has to be that which defines that difference.
   

   PS: Leela: no blushes necessary. An infinite number of objects in a finite space is counterintuitive.

   An an actually infinite amount of measurable objects or quantifiable states following one after the other is impossible. If one cannot define an actual infinite by a definite number (like 1456), than one cannot claim that an actual infinity is definable by a definite group of numbers. This is because an infinite is not a limit, you cannot complete numbers, and thus you have an indefinite number rather than an actual number defining an infinite. If it is an indefinite number, than it is a countable number, because the infinite number is potentially infinite rather than an actual complete quantity of definable numbers that add up to an infinite. When you say that there is an infinite number, you are saying that there is a definite quantifiable number of something that describes an infinite. But if there is no possible quantity of numbers that describes or adds up to an infinite, then it make no logical sense to speak of an actual infinity of numbers.
   
   To put it another way, imagine that you were a being that could actualize new beings. Starting from being 1, there is no amount of numbers that you could possibly actualize that will ever amount to an actual infinite. It doesn’t matter how many numbers you add up, you will never reach a fresh-hold that can be definitively defined as infinite. It is impossible to count from 1 to infinity. You will always have a finite countable number. This is called a potential infinite. Thus, given that it is impossible to add up an infinite procession of actualized beings, it follows necessarily, that an actual infinite cannot possibly be made up of definable parts that have been added up. Thus an actual infinite is not a quantity because there is not quantity that is infinite. You can always add one more, and if you can add one more, then you can in principle count it. But you cannot count an infinite.
   1. No amount of logically consistent additions can make an actual infinity.
   2. Therefore an actual infinite is not a group of quantifiable additions.
   Therefore an infinity necessarily transcends quantity.

 

[chrysostim83]

Hi MindOverMatter2,

I’m not sure I understand what you’ve written here, so I’ll just ask two questions.

We can see that things are larger or smaller than other things; thus there must be a smallest thing before a thing becomes nothing. It wouldn’t make any sense that a thing can be a particular size in comparison to another and yet identical in size in so far as there never being a smallest dimension to a thing. In reality, it wouldn’t really have a size, and thus there would be no logical basis for one thing being larger or smaller than another; since the words large and small would be objectively meaningless. Yet we see that things are objectively larger and smaller than other things.

Are you saying that if there were no smallest object, then there would be an infinitesimally small object?

To put it another way, imagine that you were a being that could actualize new beings. Starting from being 1, there is no amount of numbers that you could possibly actualize that will ever amount to an actual infinite.

Couldn’t God actualize an infinite number of beings all at once?

 

[Leela]

Hi MindOverMatter2,

I’m not sure I understand what you’ve written here, so I’ll just ask two questions.

Are you saying that if there were no smallest object, then there would be an infinitesimally small object?

Couldn’t God actualize an infinite number of beings all at once?

It sounds to me that by ruling out the infinitesmal from reality and saying there must be a smallest unit of size and a smallest unit of time, MindoverMatter2 is getting himself stuck in all of Zenos’ paradoxes. Don’t you think?

 

[John21652]

This is the truth: as from a fire aflame thousands of sparks come forth, even so from the Creator an infinity of beings have life and to him return again.

Cicero.

​

 

[MindOverMatter2]

Hi MindOverMatter2,
I’m not sure I understand what you’ve written here, so I’ll just ask two questions.

Are you saying that if there were no smallest object, then there would be an infinitesimally
small object?

What do you mean by an “infinitesimally small” but quantifiable form? (In respect of measurement, width, size, length)

Couldn’t God actualize an infinite number of beings all at once?

Why do you think that this would solve the problem?

What is an actual infinite number? How many numbers?

“Potential numbers” or beings that are added together cannot add up to infinity, but not because one cannot possibly count that far, but because there is no potential time or event that can transcend all numbers or countable events. Its metaphysically impossible. An an actual infinite, in order to be defined meaningfully, would have to transcend all countable numbers and thus all quantifiable events. But an infinite cannot transcend the number or its potential parts and yet at the same time be defined by potential numbers. You cannot complete potential events in the same sense that a 1000 is a completion of all the hundreds. You can always add one more and you can always count one more. All potential numbers remain potential numbers; they will only ever be potentially infinite. An actual infinity cannot be a potential infinity at the same time. Thus there is no such thing as an event that one could call “event-number-infinity.” Its not just to far to count, but rather it is impossible. If an actual infinity is the sum total of all potential events, then there is no such thing. The ideas involved in describing an actually infinite number is internally inconsistent for reasons i have labored to show above and in other posts.

In other words, the idea that an actual infinity of events can exist, is in fact a myth because there is no objective sum total that can be identified with a true infinity.

 

[chrysostim83]

Sorry, I just don’t understand any of your arguments. I might do better if you presented your proof as a numbered scheme (a list of premises followed by detailed derivations). But for now, I give up.

 

[MindOverMatter2]

Sorry, I just don’t understand any of your arguments. I might do better if you presented your proof as a numbered scheme (a list of premises followed by detailed derivations). But for now, I give up.

1. You cannot begin from one potential being or event and then actualize an infinite. A potentially infinite number of beings will never add up to infinity because you can always add one more being. Thus the number of beings and events will always and forever be a finite number that is in principle countable. This is what is called a “potential infinite”; which is different from an actual infinite number.
2. The person who claims that an actual infinite exists is also claiming (although perhaps unconsciously) that a finite number can add up to infinity, since an infinite number, in so far as it is described as a number, is in fact infinite because of the amount of potential numbers it contains. In other words, you would have to believe that potential numbers could in fact add up to an actually infinite number.
3. But the fact that a potential number, by its very nature of being finite, can never actualize an infinite, is itself (although indirectly) a disproof of the possible existence of an infinite number of potential beings (as shown in premise 1)
4. The only coherent and possible actually infinite “being” (objectively speaking) is a being that transcends the limits of potential numbers. Such a being could not be a potential number or contain potential numbers because of its infinity. Thus it would have to be timeless (without potentiality) and completely immaterial (having no potential parts), since by being so, it has no finite boundaries.

 

[chrysostim83]

Here’s what I got out of your argument. Is it accurate?

1. 
   *]An actual infinite number cannot be realized by addition by one from one potential number (premise)
   *]All numbers are realizable by addition by one from one potential number (premise, definition?)
   *]So, if there were an actual infinite number, then it would be realizable by addition by one from one potential number (by 2)
   *]Therefore, there is not an actual infinite number (contradiction with 1 and 3)
   
   It is impossible to judge this argument without a clear and distinct characterization of “potential numbers”, and a reason for thinking that addition by one is the only means of realizing a number.
   
   Anyway, if the above argument doesn’t represent your view, then there seems to be nothing I can do. Perhaps you could name some of the sources that inspired your arguments… it would be easier to figure out where you’re coming from if I knew what you were going on.

 

[MindOverMatter2]

It is impossible to judge this argument without a clear and distinct characterization of “potential numbers”, and a reason for thinking that addition by one is the only means of realizing a number.

My argument is inspired by other arguments but it is not exactly the same as other arguments and so it will only lead to more error in your understanding of my particular argument.

I saw your numbered premises and some of it is similar to mine like premise one, but i don’t think the rest fully takes into consideration the problem that i have exposed because it ignores my definition of an infinite number. Premise two doesn’t reflect what i intended, as far as i understand it. But before we get into premises; lets first define what we mean by an infinite number of quantifiable “potential beings”, and see if we both agree what that is.

When somebody tells me that they have an infinite number, i can only assume that what they mean by an infinite is that there is a definable quantity which when collectively considered adds up to an actual infinite. This is to say that an infinite is definable by the amount of numbers it contains, since we are talking about a number of something. To speak of an infinite number of beings is to admit of a distinction between each being that infinity contains.

Do you agree?

 

[chrysostim83]

When somebody tells me that they have an infinite number, i can only assume that what they mean by an infinite is that there is a definable quantity which when collectively considered adds up to an actual infinite. This is to say that an infinite is definable by the amount of numbers it contains, since we are talking about a number of something. To speak of an infinite number of beings is to admit of a distinction between each being that infinity contains.

Do you agree?

Well, I would say a number is anything set theory (i.e., mathematics) says is a number. Correspondences are used to apply this definition: for instance, to count the number of objects falling under the concept “apostle,” it is enough to exhibit a correspondence between the apostles themselves and the objects in what set theory calls 12. In the same way, if someone claims that a particular set X is infinite, it is enough for them to show that there is a correspondence between the objects of X and the objects of some particular infinite number given in set theory.

A longer answer is given by Russell in Introduction to Mathematical Philosophy. (Russell’s atheistic books should of course be avoided, but his more abstract books are usually worthwhile.)

 

[MindOverMatter2]

Well, I would say a number is anything set theory (i.e., mathematics) says is a number. Correspondences are used to apply this definition: for instance, to count the number of objects falling under the concept “apostle,” it is enough to exhibit a correspondence between the apostles themselves and the objects in what set theory calls 12. In the same way, if someone claims that a particular set X is infinite, it is enough for them to show that there is a correspondence between the objects of X and the objects of some particular infinite number given in set theory.

I am not talking specifically about “mathematics”; but rather i am talking about ontological numbers; objective quantities that exist and have quantifiable extensions that enable me individualize them. You seem to be using mathematics to define reality, but it is objective reality that is defining my use of mathematics. For example we can quantify changing reality or states; we can move from one finite state to the next and that is possible because there is not an infinite duration between those two potential states. Thus we can add them up and subjugate them to the rules of logic and addition so long as my method conforms to the objective states that i am examining; just like when we measure distance. For example there is necessarily the smallest distance between to spatial locations before they merge in to one location; and this is evidence of the ontological 1. The ontological one is a necessary expression of the metaphysical rule that a thing cannot be and not be at the same time, or that two metaphysically “different” locations cannot be the same location. I use mathematical systems such as addition only as a tool to add up ontological numbers. Okay, i grant you that set theory when considered by itself is interesting, but i don’t see its relevance when it comes to talking about an actually infinite number in the real world, since when i talk about an actually infinite number i am dealing with objectively definable quantifiable objects or states; and you seem to be avoiding my definition. So lets try one more time before we call it a day.

First of all, in terms of objective reality, is it not the case that an actual infinite “number” of potential individual quantifiable beings is infinite because of the number of objects that it contains, yes or no?

Secondly, would you not agree that you cannot reach an actual infinite from a group of potential quantifiable numbers? For instance, lets take a look at the “real apostles” of the bible. There were 12 of them. Is it not true that if you keep adding more apostles to that number you will never reach a definable limit that one can describe as actually infinite?

A longer answer is given by Russell in Introduction to Mathematical Philosophy. (Russell’s atheistic books should of course be avoided, but his more abstract books are usually worthwhile.)

I think we should deal with my definition and than ask if it is applicable to the real world. I have many of Russell books.

 

[chrysostim83]

The conversation, unfortunately, seems already to be over, since I don’t understand a word you write to me. For example, much of your nomenclature is completely unknown to me: “ontological number,” “potential number,” “actual infinite number.” The problem has been compounded for me by the fact that you seem to use “number” in such a way that the term may be freely substituted by “being” or “event.” (Note that for Russell and Frege, such a use of the word “number” would have been impossible, since, in the end, a number is merely the set of sets that fall under a particular concept.) This is why I asked about your sources, because I have never seen the words you use used in the way you use them.

One thing is clear: your concept of an actual infinite number is much stricter than mine. So it might be possible, though I doubt you would think much of the idea, that there is a one-to-one function from the set of natural numbers onto some subset of the objects of the real world (so that mathematically there would be an infinite number of physical objects) while, at the same time, this subset would not count as an actual infinite ontological number, or something like that.