Size Is Relative Because It is Measured with Respect to a Reference Frame
The size of any object, including a set, is measured with respect to a reference frame. This means that the object's size is stated in terms (units) of and is relative to the reference frame with which it's being compared. Two types of reference frame are described below. o The Reference Frame as a Ruler: Most often, the reference frame can be considered to be a ruler. That is, the size of something is stated in terms of the unit length of the ruler. For example, suppose one has an object, A, that is six inches long. If one compares A's size with a one-inch reference frame, or ruler, then A is six times as big as the reference frame or 6 inches long. But, if A's size is compared to a one-foot long reference frame, or ruler, then A is only one-half the size of the reference frame or one-half foot long. While the object's absolute size doesn't change, its size relative to the reference frame or ruler does change from six times as big to only one-half as big. o The Reference Frame as the Natural, Physiological State: This meaning of the term reference frame is best explained with an example. Suppose that a compressed spring, S, is contained within a tube containing four other, identical compressed springs positioned end-to-end. This tube of five compressed springs is S's natural or "physiological" setting. One wants to know the size of S, within this natural setting, relative to the overall tube (that is, the tube is the ruler reference frame). There are two ways of doing this. The first method would be to keep S within the tube (natural setting-reference frame) and then, as described above, compare its size with the ruler-reference frame (that is, the overall tube) and find out that S's size is one-fifth the size of the ruler within this reference frame. This method gives S's size in its natural, physiological setting and does not artifactually (that is, produced via experimental processing) alter S's size relative to that setting. A second method of measuring S's size would be to first remove it from its natural state of compression within the tube. This allows S to uncompress to, say, five times its original length. Once outside the tube (that is, in the experimentally processed setting-reference frame), one again measures the size of S relative to the ruler-reference frame of the overall tube. But here, one finds that S in its uncompressed state now has the same size as the overall tube. The difference between the two methods is that the second one removes S from its natural or physiological setting-reference frame and puts it in a new experimentally processed setting-reference frame before measuring it. This artifactually (that is, produced by experimental processing) alters its size relative to the ruler-reference frame of the overall tube. If one wants to find S's size in its natural setting, one must use the first method. As will be discussed more fully below, because the traditional method of measuring the size of an infinite subset relative to the single, parent infinite set from which it is derived entails removing the subset from the original set and creating a separate subset, this method is analogous to the second, artifactual method of measuring spring S's size in the above example. Thus, an object's size is relative not only to a ruler-reference frame but also to the natural- or experimentally processed setting-reference frame the object is in. Because the words "finite" and "infinite" both refer to size, whether or not something is finite or infinite is also, therefore, dependent on both these reference frames. One final note in this section is that most sciences consider results obtained in the physiological setting-reference frame as being more valid than those obtained in an experimentally-produced, artifact-ridden, non-physiological setting.The Traditional Method of Comparing the Size of an Infinite Subset to Its Parent Infinite Set Creates Artifacts by Its Experimental Processing Methods
Suppose that one has a single infinite set, N, of all the positive integers and that one wants to compare the total number of even integers with the total number of all integers (henceforth, the integers in N will be referred to either as Ns or as plain-integers and the even integers in N will be referred to as even-integers). Additionally, one wants to do this within the natural setting, or reference frame, which, here, is in the single set N of all the positive integers. Thus, single set N is the "experimental system" being studied. The traditional method of doing this size comparison is by splitting the evens out as a separate subset, E, and pairing off the elements of E (henceforth, the elements of the separate subset E will be called Es) with the elements of N using a function such as f(N)=E=2N. By showing that each element of E can be paired off one-to-one with an element of N, one can show that E and N are the same size. Splitting E out as a separate set and using a function to pair off each E one-to-one with an N is the experimental processing used in this method. While this method has been widely accepted for many years, it is suggested here that it is based on extensive, experimental processing, which substantially changes the situation from that in the natural setting (that is, it introduces artifacts) and, therefore, that the results it provides are questionable at best and invalid at worst. The artifacts it introduces are described next. The rationale leading to the first two artifacts is based on the fixed relationships between elements within a single infinite set. Consider an integer, Nx, in single set N. In this single set system, an inherent property of Nx is that there are also Nx-1 integers accompanying Nx within set N. This can be seen not only by inspection of the set but by the very definition of integers. An integer is defined based on the counting of elements in a sequential progression of elements (that is, a set). Without a preceding progression of three other elements, for example, a given integer, like 4, would not be defined. Therefore, if Nx is 4, this requires the presence of three accompanying integers in N (1, 2 and 3). This also implies that every even-integer in set N is accompanied by an accompanying odd-integer within the same set. For instance, 4 is accompanied by 3 in set N. Overall, within the natural millieu of single set N, there is a fixed, physiological relationship between each integer Nx and the number of its accompanying integers. Why are these relationships present in set N? They are there because all the elements are in a single set (the even-integers in set N are also members of set N at the same time), and all are marching lockstep, or in phase, towards a single, common, infinite endpoint. This single, common endpoint tethers or fixes all the elements in phase relative to one another and fixes the relationships between the number of elements present for each integer Nx. Does the traditional splitting-out and pairing-off method of infinite subset-parent set size comparison maintain the fixed relationships between integers and their accompanying elements in set N? Unfortunately, the answer is no. For example, if one splits out the integer 4 from set N into a separate subset, the 4 in the subset exists on its own; there is no requirement in the subset that there be three accompanying elements as in physiological set N. This is true for any and all integers split out from set N. Another way of looking at this is that when the even-integers in set N are split out as a separate subset E, then an E on the left side of the function f(N)=E=2N no longer "knows" about the two Ns on the right side (in totally separate and independent set N) that accompanied this E when it was in set N. What has happened is that by splitting out the even integers into subset E, the tether of the single, common infinite endpoint for the even-integers and for the plain-integers in single set N is broken, thereby totally removing the fixed relationship between an even in subset E and its accompanying integers in set N. Because they are no longer tethered to the single, common infinite endpoint with the plain-integers in set N, the Es in set E are no longer "in phase" with the plain-integers in N; they are "swinging free" and marching towards a totally separate and independent infinite endpoint in set E. Thus, the splitting out of even-integers from set N during experimental processing removes a key property of these integers in the physiological setting of the single set: the requirement that each integer be accompanied by a fixed number of other integers. This is, by definition, an artifact. It is equivalent to the one described above in which spring S was removed from its tube, allowing it to uncoil, so that its length matched the length of tube in which it was kept. An analogous situation in biochemistry would be to remove the nucleus from a cell, study it in isolation, and then assume that all the relationships of the nucleus with the rest of the cell in the physiological setting had been maintained in the isolated system. This would be an incorrect assumption. For an artifact to be a problem, it must cause different results to be obtained than would be obtained in the natural system. Unfortunately, this is what happens with the traditional method. Because they are not accompanied by their fellow elements as in set N, the infinite number of Es in subset E can be paired off one-to-one with the infinite number of Ns in set N with no elements left over. This provides the result that the size of subset E and set N are the same. However, if the physiological relationship of the accompanying integers was maintained in subset E, E would contain not only evens, it would also contain their accompanying odds. After pairing off each even in E one-to-one with the plain-integers in N, all the odds in E would still be left unpaired, thus indicating that E and N are not the same size and, in fact, that there one-half as many evens as total positive integers. The above reasoning leads to: o Artifact 1: Each integer Nx in the original, single set N is accompanied by Nx-1 other integers (that is, 4 must be accompanied by 1, 2 and 3 in order for 4 to exist within set N), but this natural, "physiological" relationship is removed in the experimentally processed reference frame of subset E. That is, in subset E, Nx is just Nx, and there is no requirement that it be accompanied by Nx-1 other elements (that is, 1, 2 and 3) within E, as in set N. and o Artifact 2: The even-integers in the physiological reference frame of single set N march "in phase" with the plain-integers towards a single, infinite endpoint, but in the experimentally processed reference frame, the Es in set E march out of phase towards a separate infinite endpoint than the plain-integers in set N. The reasoning leading to the last three artifacts is as follows. In the physiological reference frame, set N, each even-integer is also an N (that is, member of set N) or plain-integer at the same time. This "Nness" is a key, inherent property of each even-integer and, therefore, of the physiological reference frame. However, in the experimentally processed reference frame in which the evens have been duplicated and split out into a separate, and therefore independent, set, E, the evens in set E (which is the set whose size is to be measured relative to set N) are Es only. They are not Ns at the same time by the simple fact that they are no longer members of set N. They are identical to the even-integers in set N, but they are not the same element. Therefore, the inherent "Nness" of the even-integers in physiological set N has been removed from the Es in set E, by experimental processing. This is further shown by the traditional method's use of a pairing-off function such as f(N)=E=2N. In using a function, the domain (set N) and range (set E) are two different, and independent, sets. Thus, unless it's the identity function, members of the range are not also members of the domain at the same time. So, for example, while there is a "2" element in both sets E and N, any single occurrence of "2" is either in E or N but never in both at the same time, as in the physiological setting. Suppose one tries to avoid this problem by keeping the even-integers "within" physiological set N instead of splitting them out, as shown in Figure 1A. Then, as before, one would pair off, "within" the set, each N with an even-integer using the function f(N)=E=2N. A. Pairing Off "Within" Set N (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)... |__| | | | | |_____| | | |________| ... B. Experimentally Processed Reference Frame (2), 4, 6,8,10,12... Split Out Subset E 1, (2),3,4,5, 6... Original Set N Figure 1. Equivalence of pairing off "within" set N and splitting out of the evens from set N Does this do the trick of not removing the "Nness" of the Es and, thus, not altering the natural reference frame? The answer is no because, while initially, the even-integers are also Ns at the same time within this set, once the experimental processing, or pairing off, with the function is started, then an even-integer again is considered either as an E or an N but never both at the same time due to use of the function. No matter what the method used, as long as the pairing off function is used, a "2", for example, is either an E or an N but never both at the same time. Therefore, the method of size comparison in Figure 1A is equivalent to the traditional method, shown in Figure 1B. In sum, the traditional method's experimental processing technique (splitting the even-integers out as a separate set and using a function to pair off the Es and Ns) has removed from the Es one of their most inherent and defining properties, their Nness (that is., membership in set N). This leads to artifact 3: o Artifact 3: The evens (Es), whose size is to be measured, in the experimentally processed-reference frame are Es only and not Ns at the same time, whereas the even-integers in the physiological reference frame are both evens and Ns at the same time. Two related artifacts are: o Artifact 4: The experimentally processed-reference frame consists of two, separate sets, E and N, whereas there is only one set, N, in the natural reference frame. o Artifact 5: The experimentally processed-reference frame contains two occurrences of each even element, one in the separate subset, E, and one in the original set, N (Figure 1B.). While identical, these are separate, independent elements in separate, independent sets. The single set N in the natural reference frame has only one occurrence of each even integer (Figure 1A.). Overall, the traditional method of comparing the size of an infinite subset with the single infinite set from which it's derived entails extensive experimental processing, which substantially changes the original, experimental system and causes different results to be found compared to what would be found in the original system. This is by definition an experimental artifact and renders questionable the results of the traditional method.A Reinterpretation of the Pairing Off Function Can Be Used to Compare the Size of an Infinite Subset with Its Parent Set
If the splitting out and pairing off method for comparing the size of infinite subsets with their parent infinite sets creates artifacts, how can this size comparison be done in such a way as to avoid these alterations? From the above discussion, and again using the even-integers in set N as an example, any method used must avoid splitting out the even-integers into a separate subset because this breaks the fixed relationship between the even-integers and their preceding/accompanying plain-integers in set N and also removes the inherent Nness of the even-integers. One method that meets these criteria is as follows. If the Es in the split out subset E were merged back into single set N and, thus, became even-integers (in N) once again, this would mean that there would no longer be a separate subset E. All the former Es would now be back in a single set with the plain-integers in N. This means that the E on the left side of function f(N)=E=2N would truly merge with the 2N on the right side of this function. That is, each E would become a "2N" instead of an "E" or, in other words, it would become just the 2Nth N in an ordered sequence of only Ns in the single set N. If one considers each even-integer to be 2N, and not a separate E, this means that there are two N components (plain-integers) "constituting" every even-integer or, thus, that there are twice as many Ns as even-integers within physiological set N. Therefore, it is suggested here that in comparing the size of an infinite subset with its parent set, the "function" part of the traditional pairing off function (that is, the "2" in f(N)=E=2N) can be reinterpreted to represent the size of the overall set relative to the subset. This result is the intuitive approach of the layman but also appears to be the more accurate approach due to the avoidance of formation of artifacts as in the traditional method.Conclusions
In conclusion, the traditional method of comparing the size of an infinite subset with the parent infinite set from which its derived entails extensive experimental processing, which substantially changes the relationships between elements compared to those in the natural setting of the single, parent infinite set. These changes, or artifacts, lead to size comparison results that differ from those that would be obtained in the physiological setting and, thus, render questionable the results given by the traditional method. Using set N as an example, the key artifact cousing these differing results is: o each integer Nx in the physiological reference frame of set N is accompanied by Nx-1 other integers within set N (that is, 4 must be accompanied by 1, 2 and 3 in order for 4 to exist within set N), but this natural, physiological relationship is removed in the experimentally processed reference frame of subset E. That is, in subset E (whose elements are the ones to be compared in number with those in set N), Nx is just Nx, and there is no requirement that it be accompanied by Nx-1 other elements (that is, 1, 2 and 3) within E, as in set N. In place of the traditional method, another method was suggested which minimizes experimental processing and the formation of artifacts by avoiding splitting out the "subset" as a separate set. This method suggests that instead of being put in a separate subset, E, each even-integer is kept within set N, which means that it is no longer an E but just a 2N (in reference to function f(N)=E=2N). This means that there are two N components (plain-integers) in every even-integer or, thus, that there are twice as many Ns as even-integers within single set N. Therefore, this method suggests that in comparing the size of an infinite subset with the parent set from which it's derived, the "function" part of the traditional pairing off function (that is, the "2" in f(N)=E=2N) be used to represent the size of the overall set relative to the subset. This approach, while differing from the traditional method, appears to avoid the formation of artifacts. One might argue that avoidance of experimental artifacts is not required in mathematics because mathematics is abstract/mental, and not experimental, in nature. However, it is suggested here that any investigation to discover relationships about a given situation, whether done in a mathematician's mind or in a test tube, is an experiment. And, in any experiment, the need to avoid changing the situation one wants to discover relationships in should be paramount. Therefore, while the traditional method is performed in a mathematician's mind, it is still an experiment and still substantially changes the situation it is investigating as well as the results that would be obtained relative to those that would be found in the original, unchanged situation (single set N). Thus, the results it provides are still based on artifactual processing and are questionable at best and invalid at worst. Another line of reasoning that suggests that the even-integers should not be split out from the plain-integers and into a separate set utilizes some philosophy from interpretations of quantum physics. Consider set N and the relative number of its even- and plain-integers to be the subject to be observed. The mind of the mathematician is the observer and, in the traditional method, uses the splitting out of subset E and the pairing off function f(N)=E=2N as its observing instruments. After observation, the traditional method says that the observed subset E is equivalent (in size) to the unobserved even-integers in set N. However, quantum physics would disagree because according to it, every act of observation changes not only the observed subject but the entire observer-observed system. All parts of the system are intimately intertwined. If this interpretation of quantum physics is correct, then it is logically incorrect to assume that, after having removed a part (even-integers) of a real, physiological system (set N) and having put it in a separate and independent set (set E), as the traditional method of subset-set size comparison does, that this observed subset E is equivalent to the even-integers of unobserved set N. This means that in order to compare the numbers of even- and plain-integers in anywhere close to the physiological setting, the even-integers cannot be split out as a separate set and must must be kept in the same, original physiological set as they started out. One other conclusion of this paper is that the size of any object, is truly relative to the reference frame the size is measured against. This reference frame includes not only the ruler component but also the physiological or experimentally processed setting component as well. This conclusion applies not only to finite objects, such as spring S in the tube, described above, but also to infinite objects such as infinite subsets of parent infinite sets. The lessons of the spring S situation apply directly to the even-integer/plain-integer size comparison situation. In sum, the traditional method of comparing the size of an infinite subset with the parent set from which it's derived, has always assumed that the experimentally processed situation with its artifacts is entirely equivalent to the natural, single set situation. However, no other science would make this assumption without extensive testing to compare the processed and original, physiological systems. Indeed, most sciences would usually consider results based on extensive, artifact-ridden processing to be invalid. Why should mathematics be different?
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Copyright, 2001