Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2016 www.jamesrmeyer.com

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Consider the case of a pattern of a sum of ** positive numbers** such as:

** ^{1}⁄_{10} + ^{1}⁄_{100} + ^{1}⁄_{1000} + ^{1}⁄_{10000} + ^{1}⁄_{100000} +** …

In such a series, where all the numbers are positive, when calculating the limiting value of the series, it makes absolutely no difference as to the order in which the numbers come in the series. For example, the limit of the sum of the series:

** ^{1}⁄_{10} + ^{1}⁄_{100} + ^{1}⁄_{1000} + ^{1}⁄_{10000} + ^{1}⁄_{100000} + ^{1}⁄_{1000000} + ^{1}⁄_{10000000} + ^{1}⁄_{100000000} + ^{1}⁄_{1000000000} + ^{1}⁄_{10000000000} + ^{1}⁄_{1000000000000000}** …

is precisely the same as the limit of the sum of the series:

** ^{1}⁄_{100000} + ^{1}⁄_{10000} + ^{1}⁄_{1000} + ^{1}⁄_{100} + ^{1}⁄_{10} + ^{1}⁄_{10000000000} + ^{1}⁄_{1000000000} + ^{1}⁄_{100000000} + ^{1}⁄_{10000000} + ^{1}⁄_{1000000} + ^{1}⁄_{1000000000000000}** …

where, in the second case, each group of five numbers has the sequence reversed.

But we can also have patterns that include negative numbers. Such as:

(a) 1 - ** ^{1}⁄_{2} + ^{1}⁄_{3} - ^{1}⁄_{4} + ^{1}⁄_{5} - ^{1}⁄_{6} + ^{1}⁄_{7} - ^{1}⁄_{8} +** …

This is a series that has been the subject of much study, and it is called the *‘Alternating Harmonic Series’.* And it is a series that has a most remarkable property – when it comes to calculating the limit of the sum of the series, it ** does** matter in which order the terms of the series are added. So, for example, the series:

(b) 1 - ** ^{1}⁄_{2} - ^{1}⁄_{4} + ^{1}⁄_{3} - ^{1}⁄_{6} - ^{1}⁄_{8} + ^{1}⁄_{5} - ^{1}⁄_{10} - ^{1}⁄_{12} + ^{1}⁄_{7} - ^{1}⁄_{14} - ^{1}⁄_{16} +** …

which contains precisely the same terms as the series (a) above, but in a different order, has a limiting value of exactly half of the limiting value of the sum of the series (a).

This remarkable fact regarding the order of these series was proved by the distinguished mathematician, Bernhard Riemann, about 160 years ago. He proved that you can change the order of the terms of the series to give practically any value you want for the limiting sum of the series.

This is truly astonishing because here we have a result that has been known and well-established for many years without controversy, yet it is a result that is utterly irreconcilable with the conventional Platonist viewpoint. The conventional Platonist viewpoint is that all numbers ‘exist’ and that sums of limitlessly many numbers, such as in the alternating harmonic series can ‘exist’.

But how can a sum of numbers ‘exist’, but at the same time, the sum has infinitely many different values depending on the order in which the numbers are added?

The answer is simple – it can’t ‘exist’, since it is a fundamental property of addition that the order of addition is immaterial. As in the case of the Courant & Robbins contradiction, the Alternating Harmonic Series demonstrates that the Platonist viewpoint that the sum of a limitless quantity of fractions ‘exists’ is logically untenable.

For more demonstrations of contradictions arising from the Platonist beliefs in the ‘existence’ of ‘actual’ infinite sets, see The Courant & Robbins contradiction and Sums of infinitely many fractions: 2.

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There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book *There’s something about Gödel* comes under scrutiny.

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.

There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.

Flawed proofs of the Diagonal Lemma by Panu Raatikainen and Vann McGee have been added to the Diagonal Lemma web page.

16th Mar 2015 Bishops Dancing with Pixies?

23rd Feb 2015 Artificial Intelligence

31 Mar 2015 Cranks and Crackpots

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 - 2016

www.jamesrmeyer.com