Hilbert’s Tenth Problem

Goodstein Sequences

Reuben Goodstein In 1944 the mathematician Reuben Goodstein defined a set of sequences of numbers where it is not immediately apparent whether all such sequences must terminate by ending as zero. Goodstein used transfinite numbers to generate a proof that every such sequence must indeed eventually terminate.

In 1982 Kirby and Paris claimed that it is impossible to prove in Peano arithmetic that Goodstein sequences always terminate. Since then some people believe that such termination can only be proved using transfinite numbers. But there is an easy proof without transfinite numbers, see:

Proving Goodstein without transfinity

The Root Cause of the Banach-Tarski Paradox

Stefan Banach

The Banach–Tarski “theorem” claims that one solid ball is equivalent to two solid balls, each of which is the same size as the original.

Does that sounds like a load of balls?

Does it seem like saying 1 = 2 ?

For years, many mathematicians have claimed that it is not contradictory, that it only seems that way, and that it is achieved by the use of a perfectly “unobjectionable” assumption.

But a logical analysis shows that there is a fundamental logical error in the argument which completely invalidates it.

The Root Cause of the Banach-Tarski Paradox

Gödel’s Proof of Incompleteness

Kurt Gödel

In 1931 the mathematician Kurt Gödel claimed to have proved that in every possible formal mathematical system, there must be statements that can be stated in that system but which cannot be proved to be either true or false in that system.

But how rigorous is Gödel’s Proof ? Did he make some assumptions that at that time might have seemed acceptable or were overlooked, but which cannot be deemed acceptable today?

For an introduction to some questionable aspects of Gödel’s Proof see the page:

Gödel’s Incompleteness Proof

The Consciousness Myth

The notion of consciousness meaning something more than being conscious rather than unconscious only came in being less than 400 years ago.

Since then there have been interminable debates about what this notion of consciousness actually is.

There have been interminable claims about how understanding this consciousness is crucially important to understanding what it is to be human.

Some go further and suggest that there is something about human consciousness that cannot ever be replicated by any machine.

To read why the current brouhaha on consciousness is on a path to nowhere, see the page:

The Consciousness Myth

Chaitin’s “Unknowable” Number

Gregory Chaitin Chaitin’s Omega ‘number’, is a notion described by Gregory Chaitin. According to Chaitin, his definition defines an irrational number in terms of an infinite sequence of the digits 0 and 1. The nature of the definition is such that it is impossible, from this definition, to determine every digit of this number.

Chaitin claims that his definition is of great importance and that there is no possible alternative finite definition that can define all the digits of his Omega number.

But that is a fallacy that is easily shown to be completely wrong, see:

Chaitin’s “Unknowable” Number

Lebesgue Measure

Henri Lebesgue How do you assign a length to any collection of points of a line?

Henri Lebesgue’s answer was to assume that there are some points that have a width while all other points do not have any width.

For an in-depth examination of the contradictions arising from this notion, see the page:

Lebesgue Measure

And for some weird ideas regarding Lebesgue measure see the page:

Lebesgue Measure Cranks