Internal Observers
In this section, the view of an infinite set, N, relative to an internal observer is discussed. Set N is defined as being composed of an infinite number of finite-sized and discrete elements (balls, for example) relative to any location and any orientation of any element/internal observer, O, within the set. That is, wherever O is in the set and in whichever direction O is "looking", the elements of the set extend without bounds the same potentially infinite distance in all directions relative to O. O is one of the elements of N, and it can be any one of them. Further, assume that there are no pre-defined infinite subsets. Given this, then: o O sees its size as finite relative to the other finite-sized and discrete elements of the set, by definition. o O sees its size relative to the entire set as approaching, but never quite reaching, zero. It never reaches zero because no matter how far O looks, it can never see an actually infinite endpoint of the set. From O's viewpoint, the set is always potentially infinite and, thus, its size gets smaller and smaller relative to the whole set but never quite reaches zero. If O were ever able to see the whole set in its entirety, then O's size would finally reach zero relative to that whole. Luckily, for O, that can't happen! o No matter how far O travels within its reference frame (that is, the set), it will never reach the edge/boundary/"exit door" of the set due to the unending nature of infinity. o By definition of this set, relative to any element/observer, O, the other elements progress radially away from it the same potentially infinite distance in all directions. This is reminiscent of the definition of a sphere. That is, relative to any O, the potentially infinite distance radiating away from O towards a potential "edge" of the set would be the same in all directions, and, thus, O would view the set as a potentially infinite sphere. It will be an odd sphere because the observer/center point, O, can be and is at every point inside the sphere, but relative to a single O at a time, it will still appear to be a sphere. Of course, elements within the sphere can never view the edge, so that, for them, the shape of the overall set only approaches that of an infinite sphere.External Observers
Here, the view of an infinite set relative to an external observer is discussed. Again, consider set N, which was defined, above, as having an infinite number of elements relative to any location and orientation of an element/observer, O, within the set. However, now assume that there is a second observer, P, external to this set and that P's size relative to O is actually infinite (the case of a finite-sized P will be discussed below). That is, P is of the same size "scale" as the entire set N, which is also actually infinite relative to O. Therefore, P views the entire set N itself as of finite size, which means that P can "see" set N in its entirety. Then: o If P's size relative to O is actually infinite, then O's size relative to P is infinitely small or zero. Not just approaching zero, but zero itself. Relative to O, P is an actual infinite, in the big, in terms of size, so this means that O's size has finally reached zero relative to P. Because it is a part of O, O's boundary (that separates if from the other O elements), also becomes infinitely small relative to P. The O elements still exist, by definition of set N, but they are of infinitely small size relative to the actually infinite P and, thus, individual O elements and their boundaries become indiscernible to P. These elements, therefore, don't disappear (because they do exist) but instead merge into a continuous space from P's viewpoint. That is, P would observe the inside of set N as a continuous space, while O would view it as being composed of an infinite number of discrete elements. o P can see the whole or entire amount of N and, thus, can see an edge or boundary to N (wholes or entire amounts have edges or they woudn't be wholes or entireties). This is analgous to how one observes objects in daily life; when one can see the whole object, one can see the edge of the object. o P cannot "step inside" N and hope to be able to see its elements as discrete. This is because P is of a different size "scale" than the elements inside the set. P's scale is the same as that of the entire set, that is, actually infinite relative to O. P didn't get outside of N by just going a little farther within N and getting to the "door"; it got there by being of an infinite size or scale relative to any single element within the set. So, even if P tried to step inside N, it would still be infinitely big relative to the elements and would, therefore, still just see a continuous space. Thus, P is "trapped" in its reference frame (or "dimension") just as O is "trapped" in its own frame, or "dimension", inside the set. o P would be able to distinguish infinite subsets of N (if these were defined as being present) as discrete elements because these are of the same size scale as the entire set (that is, they have a finite size relative to the set) and, thus, are of the same size scale as P. o P views set N in the shape of a sphere for the following reasons. The elements in N radiate out from each element/internal observer, O, in all directions, meaning that N is a three-dimensional object and, thus, has a shape. Further, each discrete element (discrete from O's viewpoint), O, in N translates to a point in the continuous space of N as seen by P. Whatever points that P sees near the middle of this continuous space were elements that, from O's perspective, had the same size potential infinites radiating out from them in all directions. Thus, these same-sized potential infinites should translate into same-sized finite distances when seen from P's perspective. This means that, from P's viewpoint, there should be the same distance from whatever point P sees in the center of N to the edge of N, indicating that P views N as a sphere. In conclusion, relative to an infinite-sized (infinite with respect to O) observer P, set N would appear as a finite-sized sphere with an interior continuous space. An important point is that these arguments don't prove the necessity of an external observer of an infinite set; they just suggest how this observer, if it existed, would view the set. As a second case, assume that the size of the external observer, P, is finite in size relative to O. Then: o P is of the same size "scale" as O and is, thus, infinitely small relative to the actually infinite N (actually infinite from the viewpoint of an external observer). P would not just be approaching zero size but would actually be of zero size relative to N. Therefore, unless P is a member of another non-N set where it can merge into a continuous space with its own surrounding elements, P on its own would have no size and would not be able to exist independently at the size "scale" of the entirety of N. One way to view this is to consider the elements, Os, of infinite set N as forming a reference frame, size scale or dimension. Because this set/dimension is defined as being infinite relative to O, any observer, such as P, of the same size scale as O, must also exist within the set/dimension; otherwise, the set/dimension would not be infinite. An element, O, within N cannot just find the exit door and step outside to become observer P. There is no exit door and no outside in O's dimension because set N is infinite relative to O. O could travel infinitely far within N and still never hope to reach the edge of N. This is why O thinks of set N as a dimension. The only way to get outside the set is to attain either an infinitely smaller or infinitely larger "scale" or dimension than the elements within the set. This concept of dimension is discussed further below. Another analogy for a finite-sized, independently existing P would be that of a real number in the interval from zero to one "getting loose" and existing as a discrete and finite element in the interval between one and two. This is impossible, just as a finite-sized P's existence outside N is impossible. In conclusion, a finite sized (relative to O) observer P cannot exist independently outside set N.Possible Mathematics-, Physics-, and Metaphysics-Related Implications
Some potential relationships between the above ideas and mathematics, physics and metaphysics are discussed below. These relationships all refer to the main result of this paper, which is that an observer's size relative to and view of an infinite set as particulate/discrete or continuous depends on that observer's reference frame relative to the infinite set. Some of these relationships are: o The dichotomy between finite-sized internal observer O's view of set N as being composed of an infinite number of finite-sized and discrete elements and infinite-sized external observer P's view of set N as being composed of a continuous space is analogous to the distinction between integers and real numbers. That is, human observers view the set of integers as an infinite set of finite, whole chunks (corresponding to observer O's view of set N), each chunk being composed of a continuum, or continuous space, of real numbers (corresponding to observer P's view of set N). This implies that, with respect to the integers, human observers are finite-sized observers within the set of integers but that, with respect to the real numbers, human observers are infinitely-sized, external observers. It also implies that while the real numbers between zero and one, for example, seem to constitute a continuum from the human infinite-sized observer's viewpoint, if that observer could decrease his size scale to that of the real numbers, they might appear as finite-sized and discrete elements just as the integers appear in the higher size scale. Furthermore, a hypothetical external observer of infinite size would view the set of positive integers as a continuous, infinitely divisible space in the shape of a sphere. This type of reasoning also makes a certain amount of physical sense in that humans are composed of and live in a universe where everything seems to be made of finite-sized, discrete quanta or chunks (atoms, elementary particles, etc.), analogous to the composition of the set of integers. Thus, humans themselves are finite-sized, integral elements or observers, similar to observer O, within the set of the universe. Extending this reasoning, it seems possible, at least, that there may be an infinite number of discrete components within the chunks, or particles, of our universe, and that a hypothetical infinite observer would see our universe as a continuous space. From Albert Einstein's 1905 "Zur Electrodynamik bewegter Korper" article\(u)1\(d), it can be derived that there is no absolute reference frame with regard to motion within our universe; it is suggested here that this concept be extended to say that there are also no absolute reference frames in terms of size scales. o The cardinality of an infinite set is relative to the observer's reference frame. For example, within infinite set N, observer O would assign the set's cardinality as equal to that of the set of integers, .omega. However, outside set N, observer P would assign it a cardinality equal to that of the real numbers. o Brian Greene, in `!The Elegant Universe: Superstrings, Hidden Dimensions,`! `!and the Quest for the Ultimate Theory`!\(u)2\(d), notes a key difference between general relativity and quantum mechanics: Relativity suggests that space is smooth, and quantum mechanics suggests the opposite. This difference is once again analogous to the alternate views by different observers of set N (smooth relativity is analogous to P's view of set N and discrete quantum theory is analogous to O's view of set N). It also implies that both quantum theory and relativity can be thought of as different views (from different reference frames) of the same set. Both the discrete (quantum) and continuous (relativity) views are correct and only differ due to the reference frame being considered. o The particle versus wave/continuous nature of photons and all matter also appears analogous to the alternate views by observers O and P, respectively, of set N. o It is suggested that the alternate views by different observers of set N may have some implications for the process of renormalization of infinities in physics. That is, particles within human observer O's reference frame are finite in size. They are not infinitely small point particles with corresponding infinitely high energies and, therefore, do not incur the infinities needing rectification by renormalization. Infinitely small particles, relative to O, only occur in the next reference frame down from (smaller than) O and do not need to be considered as infinitely small in physical theories of human observer O's reference frame. The apparently infinitely high energies of these infinitely small particles (relative to observer O) would not be infinitely high in O's reference frame. o Infinitely small does not equal non-existent. As discussed above, infinite-sized, external observer P viewed element O's size as being zero. But, this didn't mean that O vanished into non-existence. What it did mean was that O continued to exist but merged, from P's viewpoint, into a continuous space with the other members of infinite set N. Again, size is relative to the reference frame being considered.Conclusions
In conclusion, the main result of this paper is that an observer's size relative to and view of an infinite set as particulate/discrete or continuous depends on the reference frame of that observer relative to the set. Specifically, this paper discusses how internal observer O and external observer P would view the elements of an infinite set, N. O views its size as finite relative to the other elements of the set but as approaching, but never reaching, zero relative to the size of the entire set. Because the elements radiate away from O the same potentially infinite distance in all directions, the shape of N, relative to O, approaches, but never reaches, that of an infinite sphere. O can never see the actually infinite edge or boundary of the set. In contrast, however, external observer P, whose size relative to O is infinite, views O and all the other elements as infinitely small. These elements still exist, by definition, but their sizes are zero relative to P and thus their boundaries become indiscernible. These elements, therefore, merge, from P's viewpoint, into a continuous space. P's size relative to the overall set N is finite. P views N as a finite sphere with an edge, while, as mentioned above, O views N as approaching, but never reaching, the shape of an infinite sphere without an edge. There are numerous similarities between these contrasting particulate (relative to O) versus continuous (relative to P) views of the contents of the infinite set N and integers and real numbers in mathematics and the particulate (quantum theory, particulate-like nature of photons) versus continuous (relativity, wave-like nature of photons) nature of the physical world. References 1. Albert Einstein, "Zur Electrodynamik bewegter Korper," `!Annalen der Physik`! 17, no. 4 (1905): 891-921. 2. Brian Greene, `!The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory`! (New York: Vintage Books, 1999), 129.
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Copyright, 2001