Copyright © James R Meyer 2012 - 2018 www.jamesrmeyer.com
A well-known professor claimed that my arguments (Footnote: See the appendix below Defining a set that covers the interval 0 to 1 for a brief synopsis.) regarding Platonism and Lebesgue measure as in the Courant-Robbins contradiction and Lebesgue Measure were wrong. This is the argument he used:
We define a list (Footnote: See One-to-one correspondences.) of rational numbers by taking a pre-existing list of rationals and we define that if there is a rational in the list that is close to one given specific irrational number, then that rational is pushed further down the list. The closer that rational is to that irrational number, the further it is pushed down the list. More formally the definition is:
Take any arbitrary irrational r where 0 < r < 1. Given that a1, a2, a3, … is an enumeration of the rationals in [0,1] (the interval 0 to 1, including 0 and 1), we create a new enumeration, b1, b2, b3, … where for all n, bn is the first ai whose distance from the chosen r is greater than 1⁄10n and is different from all bm for m < n (i.e., different from all bm already in the list).
While this definition is fairly informal, there is no difficulty in making it more formal, viz:
Given any arbitrary irrational r where 0 < r < 1, and any arbitrary enumeration a1, a2, a3, … of the reals over [0,1], a new enumeration b1, b2, b3, … is defined by:
∀n, ∃i (bn = ai )
if and only if
∀m, m < n, bm ≠ ai ∧ ¬[ai − 1⁄10n < r < ai + 1⁄10n ]
The professor then continued:
When we combine the definition of this list with the definition of decreasing intervals 1⁄10,, 1⁄100, 1⁄1000, … 1⁄10n, … we have a new definition.
This defines a set A*. It is impossible for r to be in the set A*. Therefore r is not in the set A* but it is an isolated point, therefore there cannot be an interval to the right or left of that point that is an open interval with a rational as its endpoint, since the endpoint would have to be r, and r is irrational.
Therefore the argument that if there were any point remaining, it would have to be within a closed interval with rational endpoints (or be a single rational) is not necessarily true.
Well, there is an obvious fallacy in the above - you cannot prove a generalization from a finite number of instances, never mind a single instance.
But there is another fallacy in the above argument, and it is much more interesting fallacy. When we analyze it, we can see that it gives us a valuable insight into how easy it is for a mathematician to be sidetracked by intuition.
When the professor says that r cannot be in the set A*, but that that it also must be a point that cannot be in the original set A that was defined (see Defining a set that covers the interval 0 to 1), what he was actually observing was that his original definition of the set A* was contradictory.
Combining the definition of the professor’s list of rationals with the definition of the intervals results in a new combined definition. That new combined definition is contradictory. On the one hand, the combined definition states that every rational is covered by an open interval with rational endpoints. On the other hand, the combined definition states that the irrational r cannot be in the set A*. But the contradiction is that this irrational r cannot be an isolated point, since to be an isolated point would require that there is an interval in the immediate neighborhood to the right side that is open with that irrational as the endpoint of that open interval (and the same applies to the left side). That is not possible under the combined definition.
What the professor has actually done is to change his combined definition from its original definition to a new definition where he deals with the contradiction by applying an additional condition - a limiting condition. He argues that since the intervals must come indefinitely close to the irrational, the limiting case is where the irrational r is not included in the set A*, but that every other point in the immediate vicinity is in the set A*.
It itself, there is no problem in specifying that the additional definition of the limiting condition is to be attached onto the original combined definition. The problems only arise when it is assumed that attaching the new condition doesn’t materially affect any other aspect of the original definition.
Specifically, in this case, the professor simply assumed that the result that he obtained - that his new definition entailed there could be a closed interval (a degenerate interval) that is a single irrational - was given by his original definition, whereas the reality is that his original definition was contradictory.
The actual non-contradictory definition that the professor produced is:
Membership of the set A* is such that either:
Note that the professor’s combined definition tells us nothing more about membership of the set A other than it is the original set A less the single irrational r - and it tells us nothing about other members of the set A.
Limits in mathematics. Why do we use them? And since we do use them, why don’t we always use them? Okay, it’s a silly question to ask why we don’t use limits to work out the value of 3 + 7. But why, for example do we use limits to work out a value when we are faced with trying to add up infinitely many numbers?
Well, the answer is really quite simple. Because there are certain mathematical situations where, unless we use limits, we either get a contradiction, or we simply don’t get an answer at all.
Take a simple example. Start with 1⁄2 then add 1⁄4 then 1⁄8 and so on. Keep doing this for ever - what sum do you get? Well, since the process continues forever, you never get to the end of the process and so you can never get an ‘end’ value. But what you can do, is to calculate the lowest value that the process of addition can never exceed. So while the additive process may never reach this value, it comes closer and closer to that value. So for this process, we can say that the limiting value is 1 (see sums of a geometric series).
You will notice that everywhere a limit is applied, somewhere a limitlessness is involved. But that does not mean that every definition that involves limitlessness requires a invoking a limit to have a meaningful mathematical value. For example, if I talk about the set of all natural numbers, it is quite clear what the value of the set is - that is, there is no ambiguity about what can be in the set, and what cannot be in the set. For example, the set of all Fibonacci numbers is defined by adding the previous two numbers, except that the first two numbers are 0 and 1:
F(0) = 0
F(1) = 1
F(n) = F(n-1) + F(n-2) for n > 1
The series begins like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
and continues without limit. We don’t need to invoke a limiting case because the elements of the set defined by the formula are perfectly well-defined as it is. And neither do we say: the set of Fibonacci numbers isn’t well defined because the definition of the set of Fibonacci numbers depends on an iterative recursion that can never actually finish - no, of course not.
First, definitions of an open and a closed interval:
An open interval is an interval that does not include the endpoints that define that interval (for example the open interval whose endpoints are 1⁄3 and 1⁄2 is the set of all points between 1⁄3 and 1⁄2 but not including the points 1⁄3 and 1⁄2).
A closed interval is an interval whose endpoints are included in the interval.
Now, define a set A in terms of ever decreasing intervals that are associated with the list:
We start with the closed interval between 0 and 1. Now take a suitable list of the rational numbers between 0 and 1 (see below). Then, going through this list of rational numbers, for the first rational we define an open interval 1⁄10 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 open interval 1⁄100 wide with that rational at the midpoint of the interval; we add those numbers to our set. For the next number, define an open interval 1⁄1000 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 1⁄10 of the length of the previous interval.
Given this definition, it is easy to show logically that this definition excludes the possibility that any point in the closed interval between 0 and 1 could not be included in some defined interval. Since each rational is the midpoint of its defined interval, then both endpoints of that interval are rationals. And since every rational is included in the listing, then each endpoint is itself the midpoint of some interval. Hence every defined interval must overlap some other defined interval - which means that the definition of the recursive decreasing intervals excludes any possibility whatever of a point that is not covered by some defined interval.
However, some people do not like this result and try to find arguments against it, such as the one described on this page.
Some people have suggested that they can circumvent the contradiction by using a list (see also One-to-one correspondences and Listing the rationals) of the rationals that are defined in terms of various conditional requirements, which render the enumeration and the sequence of intervals interdependent. Rather than trying to construct a set of rules as to which enumerations are applicable, all that is required is one specific enumeration. We can define that the set A is to be given by one specific enumeration using the pattern of rationals:
This gives a list that begins as 1⁄2, 1⁄3, 2⁄3, 1⁄4, 3⁄4, 2⁄4, 1⁄5, 2⁄5, 3⁄5, 4⁄5,1⁄6, 2⁄6, 3⁄6, 4⁄6, 5⁄6,… Note that this gives duplicates like 2⁄4, 3⁄21, but the important fact is that the enumeration includes every rational between 0 and 1 (not including 0 or 1). Clearly, there can be infinitely many similar enumerations based on similar definitions. Note that this enumeration follows a pattern that for each subsequent denominator, the values run from the lowest to the highest value of the numerator.
The enumeration can be represented by an algorithm as follows, where the calculation of the nth rational does not require the calculation of any of the rationals prior to n in the enumeration:
This can easily be made into a program as has been done here:
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There is now a new page on a contradiction in Lebesgue measure theory.
There is now a new page Halbach and Zhang’s Yablo without Gödel which analyzes the illogical assumptions used by Halbach and Zhang.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
– a page with general links:
– and a page relating specifically to the Gödel mind-machine debate:
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Copyright © James R Meyer 2012 - 2018