Logic and Language
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Copyright   James R Meyer    2012 - 2020 https://www.jamesrmeyer.com

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Understanding Limits and Infinity

Much of conventional mathematics involves contradictions that are simply ignored or dismissed as unimportant. This outlook is exemplified by a comment by John von Neumann that “…in mathematics you don’t understand things. You just get used to them. (Footnote: As quoted in The Dancing Wu Li Masters: An Overview of the New Physics (1979) by Gary Zukav, Bantam Books, p. 208, footnote. )

 

It is astonishing that anyone should be content with that approach. The goal should be to investigate the reasons for such contradictions, and by logical analysis of the reasons, to create a methodology of mathematics that eliminates those contradictions. One of the most common source of contradictions in mathematics is the failure to provide a clear logical analysis of cases that involve the notion of a summation involving infinitely many items. And in conventional mathematics a limiting condition is often invoked when it is convenient to attain a desired result, while ignoring any consideration of limiting conditions at other times.

 

Mathematicians generally assume that there is no inherent problem with the notion of an actual summation involving infinitely many items, whereas this is not the case, as discussed in Sums of infinitely many fractions: 1 and Sums of infinitely many fractions: 2.

 

Here we are going to discuss the definition of sets that involves infinitely many decreasing intervals. By analyzing such cases in terms of the limiting value of the iterative summation of items of an infinite set, the contradictions that arise from the naive notion of an actual summation of infinitely many items disappear. There is a crucial difference between:

  1. simply referring to a set of infinitely many terms (Footnote: Note, however, in some cases the elements of a set are defined in terms of a summation of infinitely many terms, and in such cases a limiting condition must be applied to the definition, as in the cases discussed here. ) and
  2. calculating a value that is defined in terms of a summation involving a set of infinitely many terms.

 

The essential details of the case that we will cover here are the same as that on the page Lebesgue measure, viz:

 

First, we define:

An open interval is an interval that does not include the endpoints that define that interval (for example the open interval whose endpoints are 13 and 12 is the set of all points between 13 and 12 but not including the points 13 and 12).

A closed interval is an interval whose endpoints are included in the interval.

 

Now we give a fairly conventional definition of a set A in terms of ever decreasing intervals:

Given the closed interval between 0 and 1 and a suitable listing/enumeration (Footnote: See One-to-one correspondences and Listing the rationals. ) that gives a unique one-to-one correspondence of natural numbers to the rational numbers between 0 and 1 (an example is given at A specific listing of rational numbers). Then, going through this list of rational numbers, for the first rational we define an associated open interval 110 wide with that rational at the midpoint of the interval; our set now includes all the numbers in that interval (not including the endpoints). For the next number, define an associated open interval 1100 wide with that rational at the midpoint of the interval; we add those numbers to our set. For the next number, define an associated open interval 11000 wide with that rational at the midpoint of the interval; we add those numbers to our set. And so on, with each subsequent open interval being 110 of the width of the previous interval.

 

The resultant contradiction

And, as it stands, this definition - which makes no consideration of limiting conditions - results in an obvious contradiction:

 

Without any inclusion of a limit condition, all the intervals that are in the set A must be open intervals of non-zero width, which means that for any points that are not in the set A, then that point would have to be either:

  1. within a non-zero width interval that is not in the set A, or
  2. a single point degenerate closed interval (Footnote: A zero width interval is also known as a degenerate interval, and is a single point, and a non-degenerate interval is a interval that consists of more than one point. Hence a degenerate closed interval is a single point that is at the same time the midpoint and both endpoints of that interval. ) that must be both a left endpoint of an open interval of A, and a right endpoint of an open interval of A.

However, option 1 is contradictory - there cannot be any non-zero width intervals that are not in the set A, since every such interval includes rationals. Hence, by the definition, any points that are not in the set A can only be single irrational points that are sandwiched between two non-zero intervals of A. But this option is also contradictory, since by the definition, every interval of A is defined as being an open interval with rational endpoints.

 

Resolution of the contradictions

While the following is intended to be a fairly informal presentation, the paper On Smith-Volterra-Cantor sets and their measure gives a more formal and detailed exposition.

 

The above definition of the set A describes a summation over infinitely many intervals that has no termination, since the intervals continually decrease while the definition does not define any state where the removed intervals or the remaining intervals become zero width degenerate single points. And it is also the case that the definition does not define a state where the endpoints of the open intervals become irrational rather than rational.

 

As noted above, the notion of an actual summation of infinitely many numbers results in contradictions, and the same applies to the notion of an actual summation of infinitely many intervals. This is because the summation of intervals is actually the addition of each right endpoint and the subtraction of each left endpoint, and so a summation of intervals is actually a combined summation and subtraction of numbers. And when there are infinitely many intervals, there are infinitely many numbers for such addition or subtraction. Attempting to define the set A without giving any consideration of limit states fails to recognize that there cannot be any actual transition from an non-zero interval that covers infinitely many points to an interval that has only one point, (Footnote: If there were such a transition, then there would be some interval of a definitive width at which such a transition occurs. ) which is why a correct analysis is impossible without the inclusion of limit states in the definition.

 

It is noteworthy that it is common in mathematics to consider the limit state of a non-terminating series or summation. For example, irrationals are definable in terms of a limit of a summation over a series of infinitely many rationals. And by an argument similar to that in Cantor’s 1874 uncountability proof, a limiting value can be defined from a sequence of rational numbers and which is a single irrational number.

 

Defining a set in terms of closed intervals

To analyze what is the correct definition of the set A, we will first consider a similar case where the intervals are defined as above, except that the intervals are closed rather than open, which means that the endpoints are included within each such interval. We will define the set described by this case as the set A*;  we define it in terms of limiting conditions, and so that the set A* is the limit state of the summation of the ever-decreasing intervals as defined above except that they are closed. (Footnote: See Formal Definitions below for the formal presentation of the definition, and you can see formal definitions of the sets described here in the paper On Smith-Volterra-Cantor sets and their measure. )

 

The limit state of a simple sequence of closed intervals of decreasing size is as a single point closed interval where the endpoints coincide and are that single point. Since there is no limit to the quantity of intervals in the definition of the set A*, there is no limit to the number of such single point limit states. But the crucial detail is that none of these limit points can be rational numbers - because if any such point was a rational number, that rational would be enumerated somewhere in the enumeration, and so it would have to be the midpoint of an interval that is, by definition, of some non-zero width, rather than an interval of zero width. Hence each such limit point can only be an irrational point where every other rational number is either smaller or larger than it.

 

This may appear to be counter-intuitive, but history shows us that we should never set logic subservient to intuition. It can be seen that as the enumeration progresses, the overall situation is that the denominator of the associated rationals becomes larger and larger. The limit state is that the denominator is infinitely large, in which case the number cannot be a rational, since all rationals have finite denominators hence the limit state is an irrational number. (Footnote: If we consider the decimal representation of a rational, it is as a decimal point followed by either a finite sequence of decimal digits, or an infinitely long sequence where there is at some point a finite sequence of digits that repeats itself indefinitely. It can be seen that as the enumeration progresses, there will be rationals with longer and longer such repeating finite sequences. Hence the limit state is as infinitely long sequences that do not repeat, and such numbers are irrational numbers. )

 

Given that this is the case, then it follows that the limit state of the set A* includes all these limit state irrational single points as well as all the defined non-zero width intervals. Conversely, the remaining intervals are ever-decreasing open intervals, and the limit state of ever-decreasing open intervals is also when the endpoints coincide, the difference being that for open intervals, the intervals disappear completely, so that the limit state for ever-decreasing open intervals is that limit state intervals do not exist, although the positions of such limit states do exist. (Footnote: While the term degenerate interval is normally considered to be a single point closed interval, it might be better to use the term “degenerate closed interval” to denote a single point closed interval and the term “degenerate open interval” to denote the position of an open interval whose left-side endpoint and right-side endpoint are coincident. )

 

We can now see that the limit states of the removed intervals and the limit states of the remaining intervals are logically compatible and that both lead to the same result for the constitution of the set A*, so that the formulation of the definition with an explicit inclusion of the limit states gives a non-contradictory definition of the set A* as the complete interval between 0 and 1.

 

Defining a set in terms of open intervals

We can now give a logically coherent definition for the set A for open intervals rather than closed intervals, and we define it in terms of limiting conditions, and the limit state of the ever-decreasing open intervals is that the intervals disappear. And conversely, the remaining intervals are closed intervals, and the limit state of ever-decreasing closed intervals, as described above, is as single point degenerate intervals where the endpoints coincide. In the same way as in the case above, these limit points cannot be rational, and must be irrational, and they are the very same limit points as for the case above.

 

Hence the set A, when properly defined, can be seen to be the interval between 0 and 1 except for these limit points. We can call the set of these limit points the set B.

 

These points are not denumerable - it is not possible to define a one-to-one correspondence from natural numbers to all such points. The reason for this is not, as is commonly believed, because that there is in some sense “more” points in the set B than the set A; it is simply because the limit state is never actually reached at any point in the continuing decrease of the width of the intervals; there is no definable transition from the infinitely many decreasing intervals to the infinitely many limit state points. (Footnote: Note the difference between these cases which involve infinitely many limit states and simple single limit cases such as defining a irrational number as the limit of a summation of a single series. )

 

It was demonstrated on the page Lebesgue measure that every such limit point is uniquely associated with a “complete interval ” (Footnote: We use the term “complete interval ” to indicate an interval that is not a sub-interval of any defined interval of A apart from itself. ) of the set A. And from that it follows that there cannot in any sense be “more” points in the set B than the set A, and since that is the case, then the total width of these zero-width points of B cannot possibly be greater than the total width of the associated non-zero width intervals of A.

 

The fallacy of the conventional notion that the Lebesgue measure “proves” that the set A must be a total width of less than 19 is now starkly apparent. When one considers that for the closed interval case, a logical consideration of the limit states of the set A* indicates that this set is the entire interval between 0 and 1, and yet, according to the conventional stance based on Lebesgue measure theory, the measure of this set A* must be less than 19 , a conclusion that is patently absurd.

 

Refusing to accept the logic of limit states

Many people appear to have an emotional attachment that drives them to attempt to defend the notion that the set of numbers in the set B constitute a “bigger” infinity of numbers than not only rational numbers, but also a “bigger” infinity than the set of all numbers that are in the set A. These notions of different sizes of infinity originated in religious beliefs and which have never been proved, despite the fact that many theorists have spent their entire working lives working with that belief, see Proof of more Real numbers than Natural numbers? and also Cantor’s religious beliefs, Cantor’s invention of transfinite numbers and Why do people believe weird things?

 

These people feel that they must automatically reject any suggestion that there might be a logical flaw in their belief system, rather than attempt to analyze it logically. Their emotional attachment to their set of beliefs also means that, even though all the points in the set B are all single zero width isolated points, they must believe that the total width of such points must be greater than the total width of the set A, and their belief in the “bigger” size of the infinity of irrational numbers than rational numbers leads them to believe that this “bigger” infinity can in some way account for the claimed difference in measure. But logical analysis show that this cannot be the case, as shown on the page Lebesgue measure, and in the paper On Smith-Volterra-Cantor sets and their measure.

 

While these beliefs have a superficial veneer of coherence, a detailed logical examination shows up the major flaws in the overall system that is the conglomeration of the individual concepts. It is rather like observing a completed jigsaw from a distance, where at first it may appear to be satisfactory, but as one approaches closer, one can see that several of the pieces have been put into the wrong places.

 

In order to defend their beliefs, some people cling to the claim that it is not imperative to apply limit states, but as demonstrated above, this results in a blatant contradiction. But it appears that many people hold stubbornly to their belief systems, which is why one sees arguments that strenuously attempt to justify the arcane notions of Lebesgue measure. But why do people that proclaim themselves to be logicians who prioritize logic over emotion continue to defend what is logically indefensible?

 

Hopefully at some point in time, mankind will put aside primitive illogical concepts that are long past their sell-by date, and instead strive to develop systems of mental concepts that are fully coherent and which do not result in contradictions. Systems of mathematical concepts and logical concepts that are free of contradictions are known as consistent systems, and I believe that it is only by applying rigorous logical analysis that the contradictions that are evident in conventional mathematics will be eliminated.

 

For a more detailed overview of the above, please see the paper On Smith-Volterra-Cantor sets and their measure.

 

Formal set definitions

As noted above, since the definition must include the notion of a summation that is to be applied to an infinite set (by a variable n whose domain is infinite), then as noted in Sums of infinitely many fractions: 1 and Sums of infinitely many fractions: 2, an infinite summation of fractions is a contradictory notion. Hence to make a well-defined definition of the set, such a definition of the membership of A should include the application of a limiting condition. This gives the definition of the set A as:

 

A   =  Limit of Union  {r ∈ R : q(n) – 1/(2·10n) < r < q(n) + 1/(2·10n)}

 

where q(n) is a function that lists the rational numbers between 0 and 1, defined for n > 0.

 

The set A* is similarly defined as:

 

A*  =  Limit of Union  {r ∈ R : q(n) – 1/(2·10n) ≤ r ≤ q(n) + 1/(2·10n)}

 

Page last updated 04 Feb 2021

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Copyright   James R Meyer   2012 - 2020
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