Sums of infinitely many fractions: 1

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.

And more …

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.

Peter Smith’s ‘Proof’

It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.

There’s something about Gödel by Francesco Berto

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.

Easy Footnotes

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).

O’Connor’s “computer checked” proof

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.

New page on Chaitin’s Constant

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.