Examples of How to Imagine the Infinite.

Robert A. Herrmann, Ph.D.
25 JUL 2013. Latest Revision 24 FED 2017.

Examples of How to Imagine the Infinite.

Some people who want to pollute our world with some rejected late nineteenth century concepts claim that what I present here does not exist. They insist that human beings are like them. They insist that human beings are incapable of imagining the "infinite." This claim is utterly false.

(Note: At the end of this article, I have added any additional article that appears on this website on the "infinite" and non-secular mathematics.) What does the term "completed infinite" mean? As you'll see shortly, the notion of the "potential infinite" is rather easy to imagine. A completed infinite would be "something" that "contains" a potentially infinite "something." So, how do I mentally image a completed infinite? As you will notice from the following descriptions, I cannot successfully draw this mental image on a piece of paper. But, it is shown that properly described mental images will coincide with the concept of the (completed) infinite.

The concept of the observational finite is physically established by considering the number of photons that can enter a human eye over a life-time, photons that provide the sensation of "sight." A life-time is an atomic-clock counted span of time, where the counting starts and ceases at a particular numerical value. It is acknowledged that such "sight" and counting can be physically modeled by a photo-multiplier in place of the human eye, an instrument that registers the reception of photons via a counting mechanism, and a life-span is modeled by a fixed duration for the machine's operation.

Mathematically, this notion of the finite is "conceptually" extended by allowing an extension of the observational finite to that of a counting notion that can continue. It is finite counting notion that intuitively "does not cease." Hence, the "life-span" concept of man or machine is removed. Historically, whether one needs to "imagine" such a process that has no human observational evidence or simply accepts this finite concept based upon some "unknown" method of comprehension is not generally addressed. For many fundamental mathematics processes, descriptions are given in terms of common and well understood language elements. This is part of the very important "intuitive" aspects of mathematics.

An Infinite White Road; the Beginning of Various Completed Infinities.

Close your eyes or go into a "completely" dark room. That is, in some way, remove the visible light from your view. Now imagine (mentally image) a white road with an assumed fixed width beginning at a position within this black background. There is a straight line segment where the road "starts" in the black background. The entire road is totally surrounded by the totally black background. The edges of the road are straight lines and, from your mental view, the road appears to extend towards the "upper right." The road appears to be slowly dwindling in width until it appears to be just one single dot. This is a "vanishing point." However, you still known that the road has a fixed width. From your experience with actual roads, you conclude that the road extends further and further from its start position. This is the potential(ly) infinite concept - a step-by-step process that "continues." Now the background has no "visible" boundary. (This fact is what yields a meaningful vanishing point.) But, notice that even for the unbounded black background, the road still extends in a direction relative to your mode of viewing. No matter how you imagine this dwindling white road, it is "suspended" within the black background.

What actual procedure has been done to invoke these and the other images to be described later?

A description is presented in linguistic form that, via the concept of "mental intentions," should be, at least, mentally perceivable. There may be some individuals who do not have the mental capacity to translate the description into a mental image. One slowly reads the words and then applies the process. The words have meaning based upon previous human experiences. When I do this, I am mentally aware of a mental voice, the "speaking to myself voice." Images are mentally produced. They are not of the same character as they would be if I were viewing with my vision. Indeed, for me, they are rather not related to my vision since I can superimposed various ordinary images over my visual images without any interference.

I now command images to appear in a place where I have little or no experience; that is, within a completely black region. It is a situation that does not correspond to any physical laws with which I have had experiences or know about. Images appear in the forms I demand. I have not given any further description as to how my mental intention is to produce the desired results. The requirement is that what I read or "mentally speak" is to be carried out. I can "suddenly" perceive mentally a completed road. In all of the presented examples, the vanishing point is not the point of concentration. It is the "entire" road that is being mentally conceived as viewed from the background. It is not being produced in a step-by-step manner. When I make this additional "observation," this yields a completed infinity that corresponds to my mental intention. The images mentally appear as I have requested. This "request" is a deterministic notion, but it determines or produces the complete outcome without an algorithm. That is, without a defined process that produces the images.

This form of imaging is not related to an external brain probe that can produce mental images. This form is an internal "command" or "an internally stated intention" that produces the images and the final result. According to Eccles and Robinson (1984) and others, if I intended to perform certain actual repeated physical motions, then there is no physical connection found between certain mental intentions and the actions performed. But, in this case, the actions are all related to internal mental actions that yield internal mental results. It is of little significance if there is speculation that a "such and such" describe physical process "could" produce, from a type of mental command, such mental actions - the forming of mental images. Only predictive and observable experimental results based upon stated physical laws can overcome statements like "there is no physical connection" or "that no actual physical processes have been shown to exist that predict these mental outcomes." Indirect verification for assumed entities and their behavior only implies a possibility and does not indicate fact.

As mentioned, the potential infinite is a concept relative to the step-by-step notion, step-by-step actions, instructions that are carried out in a step-by-step manner. It is a myopic restricted local view with an added requirement. As to operations with numbers, it states that one can apply operations only in a step-by-step manner and only applies to the "intuitive" idea of the finite with an added requirement that goes slightly beyond the previously described finite. Further, it can correspond to a specific axiom system that is defined only in terms of operations performed on finite collections of objects with what is comprehended as an added result. But, this implied result is NOT usually stated. You tend not look at the results of the operations as a total collection.

You start at the beginning of a road. You are on it and walking. Physically, the potential infinite is the idea that you can take one step after another without a time boundary. All you know is that you can, with this restricted view, continue to take that step. Indeed, you are constructing an imagined white road since at each step the portion of the road upon which you step simply appears and is adjoined to the previous portions. You intuitively never cease this view. The distance you have covered is finite and you can add to this distance in a step-by-step manner. Although, you continue to walk, you give no thought as to ever stopping. There is intuitively no termination to this road construction process. You are "extending" the previous finite in this step-by-step manner.

Various mathematically axioms imply the previous description. This implication comes from mathematical logic where the finite notion of the step-by-step construction of a formal proof is employed without further discussion as to what the term "finite" signifies. Such axioms may be stated in operational form such as given an object denoted by x, then there exists an object denoted by x′ and x is not equal to (not considered the same as) x′. The axioms allow one to "write" an array of symbols x′, (x′)′, ((x′)′)′ . . . . It is the concept of the ". . ." that is employed. This means that one needs, at least, to mentally conceive of repeating this "priming" operator "without termination" although the axioms do not make this additional claim. This mathematical idea also carries the name of the potential infinite. It is the implied "without termination" that is the added requirement.

Returning to the original image, unfortunately, there is one aspect I cannot draw. In my mental image, there is no edge to this black background. There is such an edge controlled by an individual's field of vision when physically "seeing" any collection of objects such as an actual road moving off to the upper right in ones visual-field. Further, any finite drawing must have an edge. So, what part of this description is the completed infinite?

It is created from your view that there is an unbounded black "something" that contains - that surrounds - the entire road. It is your unbounded view that is a mental model that yields the complete road. You are, indeed, viewing the completed road. One might claim that there is "nothing" in the black background. But there is. Your viewing stance is in it. It's from this background that you are viewing the entire road. From this view, the road is, from this imagined view, complete. In such cases, you can't rationally view yourself as part of the image. For if you could, then this leads to a logical regress and your brain would close down.

[The logical regress occurs this way. If I can image myself viewing this scene, then I can imagine myself imagining myself viewing this scene. Then I can image myself imaging myself imaging myself viewing this scene. Etc.]

Imagining more Complex Completed Infinities.

Close your eyes or go into a "completely" dark room. That is, in some way, remove the visible light from your view. Imagine a glass or other type of square panel or plate that is placed perpendicular to the road and has the same width as the road. The panels are vertical and placed one behind the other but with a fixed distance between them. This is done so that, from your view of the panels, you can "see" a slim rectangular region across the top of each panel. You only see the first panel completely. For the other panels, you see the top and the "right-hand" side. The panels do not "actually" change in width but retain the same width as that of the road. The space between them "appears" to be slowly dwindling as the panels appear to get less wide. From your experience in viewing such events, this is what you expect from a view of this configuration if it takes place from various positions within the view itself. Thus, the panels "appear" to get smaller and smaller in size until they appear to be just one single dot. From your mental view, they appear in a row that extends towards the "upper right." They extend towards a "vanishing point."

The fact that the background has no boundary and the black surrounds this row of panels is what, as described below, yields a meaningful vanishing point. (You might have to color the panels to imagine this successfully.) Although this black background has no "visible" boundary, you still have a direction relative to your mode of viewing.

As you start from the closest panel and move mentally to the mentally appearing smaller and smaller panels, you mentally notice that there is another panel "after" the one you stopped at. For this case, this is the notion of a potential infinite collection of panels.

As before, there is one aspect of the image I cannot draw. This is the actual aspect that, for me, yields a model for a completed infinite collection of panels. In my mental image, there is no edge to this black background. Nothing is there that seems to stop the panels. There is such an edge controlled by an individual's field of vision when physically "seeing" any collection of objects moving off in the same visual direction. Further, any finite drawing must have an edge. So, what part of this description is "the panel" that includes all the other panels?

But significantly, I can mentally view the entire collection of panels "suddenly" as a complete entity. You are viewing the collection from where? It is within this "view" that each of the panels exist. It is within this "view" that these images occur. All the panels mentally exist within this view and this "complete" view can be defined as "something" that certainly contains each panel. So, what do you have? If this "view" did not mentally exist, then what are you mentally imagining? Using this obtained complete mental view, one has a model for the completed infinite. Notice that the panels can be on the white road OR you can mentally create the view without being guided by a white road. Thus, the black background can be thought of as the "something" from which you are viewing the entire road, the entire road and panels or only the collection panels as a complete object. From the black background view, my imagined panels merge with the vanishing point, where the mental image does not distinguish between them.

For the potential infinite, except for the first panel, as you view a particlar panel, then there are those that "came" before the one you are viewing. A panel and the finite collection of the panels or objects that came before is also a way to view the potential infinite.

Here are objects that yield a third and forth infinity that is controlled by the original one. Consider a finite set of rules that you can follow that produce an object. You follow the same rules each time an object is produced and attach the object to a panel. Of course, the objects attached to the panels could be identical. Indeed, each identical object can have a distinct "numerical" name that corresponds to a panel name, like the Kleene tick marks ||||. The "front" of a panel is defined as the surface you can mentally perceive from the start line if finitely many of the "previous" panels are removed. (Notice how the items are given in terms of finite stuff that exists or finite collections that have been produced.) You then either use what you have constructed thus far to construct a new panel object or apply a fixed rule relative to or not relative to the other panels to construct an object on a panel. So, construct the objects one-at-a-time and glue the first object to the "front" of the first surface. Then as the objects are completed, glue them to each successive panel. In this imagined scenario only sequential behavior is being considered. There are no "time" considerations. Each panel has an object glued to it.

Notice that I have not given an algorithm, a finite description, for any processes that yield the "view" from the written description. I base this result only upon a form of "observation." The experimental aspects are the same as the interpretations given to physical observations by a human being from personal observations or observations of a machines output. Thus, in order to establish that there is a common mental process going on that yields the stated results, one requires a preponderance of evidence that others can also imagine this same view. As mentioned, it is of little significance if there is vast speculation that "such and such" described physical processes "could" produce such actions or such images. How these imagines may be produced is of no significance relative to their comprehended mental appearance as produced by the corresponding linguistic description.

It seems that, from the previous description, I can produced "suddenly" the entire mental model for the class of glued objects; a completed infinite image of glued objects. Is there a fourth completed infinite that can be viewed relative to the objects and panels? Use the road and panel image. Just write on the road immediately in front of the panel (between the panels after the first one) the "number" for the constructed object. (Shortly the construction of such "numbers" are set-theoretically described.) Finite collections of symbols, written left-to-right, with repetition can be imagined and each corresponds to unique natural numbers. Of course, if you can imagine this, you can also make finitely long rows of such symbols of road width length but piled one on top of each other. This is but a potential infinite naming process. Once again one might be able to mentally view all of this as "suddenly" appearing and have another example of a mentally viewed completed infinite. This gives, at least, four examples; the road, the panels, the glued objects, and finite and yet distinct sets of symbols.

In order to avoid a phrase such as "continual creation," there are secular cosmologies and, indeed, it may soon be announced that such a cosmology is necessary, that require a completed infinite universe in terms of "space" or space and physical material. At the least from the viewpoint of quantum field theory, an infinite space and material universe requires infinite quantum fields. One does not find a description within quantum field theory for a mentally imagined quantum field. In the 1960s, they were simply called "immaterial things" that are endowed with properties. But, at least, one of the "infinite" ones can be imagined. For an infinite collection of finitely large slices of a universe thought of as panels, a quantum field can be considered as the black background.

There is, however, one more highly significant view. Once the panels and any objects that may be attached "appear" in your view, then the white road can be removed. This aids one in imagining the panel only view. The sudden view of the panels and any objects on them continues to be a view of a completed infinite.

Of course, since modern mathematics is based upon axioms that need not be related to physical reality, then this model may be considered unnecessary by a "purest." But, it is related to establishing that a major collection of axiom is consistent.

Imagining the Complete Infinite Set of Natural Numbers -
A major Aspect for a Mental Model for the Consistency of Set Theory.

(This is a somewhat more technical discussion.) Consider each panel as made of glass and call it a "platform" or "construction stage." "A set x is formed by choosing the sets which are to be members of x . . . . there is a stage after all the stages in S provided we can image a situation in which all the stages in S have been completed." This is what Shoenfield writes in [3]. What is he writing about? He is attempting to show the formal axioms of modern set theory have a concrete model and, hence, are true relative to the notion of a concrete model. Now substitute the term "platform" for "stage." The platforms and our panels will coincide.

To "form" means to either place sets defined as symbols between the symbols { and } or imagine that the choosing process can be continued in a specific manner. For example, consider the O as the "empty set." This is defined as a set, where there are no sets used to form O. There is a relation between sets called the "elementhood" relation. We use the phrase "a is a member of or element of A" to indicate that a set "a" is so related to the set A. But, for O there are no other sets that are so related to set O. Please notice that I use the word "set" as a general name for an object being discussed in "set theory." Terms like "collection" or "class" and the like are not part of the formal language of the axioms of modern set theory. They are used for other purposes, however.

So, the set O has no members that are considered as sets. It is the only set not formed this way. All other entities must be formed from previously formed entities. Indeed, one thinks of these as merely constructed symbolic forms.

"Sets" are considered as formed on the platforms. For all but the first platform of formation where O resides, that is for each platform upon which a "set" is formed and not just declared to be a "set," there are finitely many platforms that intuitively come before or are previous in our construction. (One can think of such platforms as glass platforms through which one can see the other platforms with the formed constructed sets sitting on them.) Now you can form at such platform S, constructed sets composed of collections of constructed sets formed on platforms that are intuitively before the formation platform S. (The formed symbol O is not a constructed set but is a model for the empty set.) Note again that there are only finitely many previous platforms.

We use the "union" operation. For two sets A and B, then A U B is composed of the sets that are in A or in B. We start the first platform with the set denoted by the symbol O and considered it formed at the first starting platform. All other formed entities are constructed sets. Now for the next formation platform S, we form the constructed set O U {O} by choosing symbol O. Since in this special case, O has no members, then it contributes nothing to the O U {O}. So, all we have is the one declared set O that is in {O}. Thus O U {O} = {O}. Now for the next platform, we form {O} U {{O}} = {O,{O}}. Next we have {O,{O}} U {{O,{O}}} = {O,{O},{O,{O}}}. Notice that this symbolic form, this constructed set, can also be formed by taking the entities on the previous platforms O and {O} as elements of the next construction set and including the additional entity you get by enclosing these previous "sets" within { and }. Indeed, after the O and {O} are formed, in each case, the imagined pattern for constructed sets is

{previous platform sets as elements, {previous platform sets as elements}}

Thus the next platform has on it the string of symbols

{O, {O},{O,{O}},{O,{O},{O,{O}}}}

and this equals {O,{O},{O,{O}}} U {{O,{O},{O,{O}}}}.

Notationally, we need to identify the formed constructed sets at each platform of formation. We do this with "tick" marks. Tick marks, like x|||, can be used as indicators for entities constructed at various platforms, S|||. Intuitively tick marks can be counted and the number of them used as an abbreviations for a finite collection of these tick marks. We now make use of the symbols used, since childhood, that describe the counting of the tick marks. Except for the notion of this learned counting, no other properties are given to these numbers. Then put O = y0. Thinking of O as a box, there is nothing in it to count. The methods used within Mathematical Logic allow one to analyze these forms relative to the symbols employed and their location relative to their order as represented by a finite sequence of ordered members of informal natural numbers [2, p. 12, Third note]).

Let constructed set y1 = {O} be a platform formation, where y1 contains one member. Thus you can count how many we have so far if you wish. Then the process is started as indicated above. There is a platform formation that is composed of the members of y1 and the platform formation y1 = {O} itself. This yields the constructed set y2 = {O, {O}}. As above, y2 is formed from two previous entities that are formed at two previous platforms, the forms O and {O}. Using counting numbers, we have the three platforms (stages) S0, S1 and S2. Continuing to follow the simple rule from here out, gives for a stage S3, y3 = {O, {O}, {O, {O}}}.

Clearly, we cannot continue this specific symbolic method if we were "writing down" strings of symbols due to physical and time constraints. This is why we do it mentally via the imagination. Thus, we employ an informal induction definition. For the third stage on, let yn be formed at platform Sn. Then yn+1 is formed at a later platform Sn+1 by using the finitely many members of yn and yn itself (i.e. { yn}). (We have also inductively defined the construction platforms.) In general, the constructed sets produced might be considered as satisfying the notion of the potential infinite. Those that claim that we can't imagine the completed infinite stop with the potential infinite, the increasing large finite constructed set idea with an included "without termination" intuitive notion.

What can the forms O, {O}, {O, {O}} etc. represent in set theory? I have used the "informal" symbols 1, 2, 3, etc. as abbreviations for tick marks and they are only relate to "counting." They should not be considered as related to natural number arithmetic (1 + 2 = 3) or the order (0 <1 < 2 < 3 < . . .). Here is where a notational change needs to be made. But, it is not usually done. It is often to be "understood."

What can be done in the set theory in [3] is to define a formal natural number as a symbolic form 0 to be the declared set O, then 1 is {O}, 2 is {O,{O}}. The symbol 0 can be considered as an abbreviation for the "symbol" O. (In the subject Mathematical Logic strings of constructed symbols are investigated.) Given any formed set A. Then B is a "subset" of A if each member of B is a member of A. Now, as a model for set theory, the empty set O is a subset of any set A. Why? Well, the statement that "every member of O is a member of A" holds since it has no "members." Hence, O is a subset of {O} and not equal to {O}, {O} is a subset of {O,{O}} and not equal to {O,{O}} etc. The set O is also a subset of {O,{O}}.

Now define, for these "numbers," an order < by the concept of "subset." Let's see if this behaves like what we are use to. We get that 0 < 1 < 2, etc. and 0 < 2 etc. Denote by N this compete imagined stuff on the white road panels. These panels can have formed O or the finite constructed sets formed from the O and { } glued to them, or the 0, 1, 2 etc. simple symbolic abbreviations for them. The term "set" is being modeled and here means a collection as informally understood and, hence, the imagined N is simply declared to be such a "set." Then N models the Axiom of Infinity in [3. p. 326].

In all of the above cases, a higher-intelligence "reversed view" can be described. There is the black background "beyond" the vanishing point. Thus, one might simply note that it seems reasonable that "something" could view the white road as well as the various constructions from this "far" black background towards the beginning of the road. That is, view everything in the reverse direction, so to speak. I am not convinced that any human being is able to take this view.

We now come to the conceptual part of set theory as discussed by Shoenfield and our imagination [3. p. 323-324]. Each of the constructed sets is intuitively finite and each is specifically defined. But our constructed sets have an additional property, they are formed at some platform. We now need to consider a question. You have the platforms at which constructed sets are formed. These platforms do not model set theory sets. Does there exist a Shoenfield platform S, where each of the platforms used to construct these forms is "before" S? A platform is but a method for construction or a "view." Shoenfield simply states that maybe we can image that such a platform S exists for the collection of all these inductively defined constructed sets. He doesn't indicate that this is actually possible, however, and this "maybe we can" is what is rejected by many. Thus, assuming we can so imagine S, then the declared set N of all of the inductively defined constructed sets yn is "on" or "in" S, where N is a completed infinite declared set.

Can the S platform actually be described? Yes. It is black background that is the platform from which one is viewing the imagined construction. From this viewing platform, one can mentally erase the panels upon which the stuff is glued and retain, in the view, the entire collection N itself as if it is floating above the white road.

This is an important additional discussion on the infinite and non-secular mathematics.

[1] Eccles, J. and D. N. Robinson, 1984. The Wonders of Being Human Our Brain and Our Mind, The Free Press, New York.

[2] Mendelson, E., (1987), Introduction to Mathematical Logic, Wadsworth & Brooks/Cole Advanced Book & Software, Monterey, CA.

[3] Shoenfield, J. R., 1977. The Axioms of Set Theory, In: Handbook of Mathematical Logic, (ed. Barwise), North Holland, New York, pp. 321-344.


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