Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2018 www.jamesrmeyer.com

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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 *a*_{1}, *a*_{2}, *a*_{3}, … is an enumeration of the rationals in [0,1] (the interval 0 to 1, including 0 and 1), we create a new enumeration, *b*_{1}, *b*_{2}, *b*_{3}, … where for all *n*, *b _{n}* is the first

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 *a*_{1}, *a*_{2}, *a*_{3}, … of the reals over [0,1], a new enumeration *b*_{1}, *b*_{2}, *b*_{3}, … is defined by:

∀*n*, ∃*i* (*b _{n} = a_{i}* )

if and only if

∀*m, m < n, b _{m} ≠ a_{i}* ∧ ¬[

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}⁄_{10}^{n}, … 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

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

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:

- each rational is covered by an open interval with rational endpoints

or *r*is not in the set, and there is a limiting condition such that all points in the immediate neighborhood of*r*are in the set**A***.

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

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.

Footnotes:

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

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

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

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:

^{1}⁄_{2} |
^{1}⁄_{3} |
^{1}⁄_{4} |
^{1}⁄_{5} |
^{1}⁄_{6} |
… |

^{2}⁄_{3} |
^{2}⁄_{4} |
^{2}⁄_{5} |
^{2}⁄_{6} |
… | |

^{3}⁄_{4} |
^{3}⁄_{5} |
^{3}⁄_{6} |
… | ||

^{4}⁄_{5} |
^{4}⁄_{6} |
… | |||

^{5}⁄_{6} |
… |

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

The enumeration can be represented by an algorithm as follows, where the calculation of the *n*^{th} rational does not require the calculation of any of the rationals prior to *n* in the enumeration:

- Let
*t*= 0 - If √ is a Natural number, then:
*m*= (−1 + √ )/2 and the*n*^{th}rational is (*m*−*t*)/(*m*+ 1)- Otherwise let
*t*=*t*+ 1 and repeat from step 2.

This can easily be made into a program as has been done here:

Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

How you can tell if someone is a crackpot

A review of Buldt’s *The Scope of Gödel’s First Incompleteness Theorem*

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.

8 Apr 2016 Are we alone in the Universe?

13 May 2015 Good Math, Bad Math?

31 Mar 2015 Cranks and Crackpots

16th Mar 2015 Bishops Dancing with Pixies?

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

www.jamesrmeyer.com