Logic and
Language


Copyright   James R Meyer    2012 - 2025 https://jamesrmeyer.com

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

Goodstein Sequences

 

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

Chaitin’s “Unknowable” Number

 

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

The Axiom of Choice

 

Why would anyone rely on an assumption in an argument when it leads directly to a contradiction?

 

The “Axiom of Choice” is an assumption used by some mathematicians, and they justify the assumption by claiming that it gives certain results that they want to achieve.

 

But it also leads to the result that one sphere is equivalent to two spheres which are each of the same volume as the single sphere.

 

Which might be a result that someone wants if they want to engage in fantasy, but it certainly isn’t what a scientist would want from his mathematics.

 

For a closer look at the “Axiom of Choice”, see the page:

 

The Axiom of Choice

Hilbert’s Tenth Problem

 

In 1900 the mathematician David Hilbert posed 23 major problems that were at that time all unanswered. Problem 10 was the question as to whether there can be a finite process which can definitively tell whether there are natural number solutions to a certain type of equation known as a Diophantine equation.

 

In 1970 Yuri Matiyasevich claimed to have proved that the answer to Hilbert’s question was that it is impossible for there to be any such process. But is Matiyasevich’s proof rock-solid? See the page:

 

Has Hilbert’s Tenth Problem really been answered?

Lebesgue Measure

 

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 regrading Lebesgue measure see the page:

 

Lebesgue Measure Cranks

Indefinable Numbers

 

In 1905 Julius König⁠ suggested that there could be numbers that exist but which are impossible to define.

 

An indefinable number might be said to be a number that contains an infinite amount of information that cannot be summarized by any finite definition.

 

But Georg Cantor, the founder of the theory from which König’s idea of indefinable numbers arose, found the notion deeply disturbing when it was postulated, stating that ‘Infinite definitions (which are not possible in finite time) are absurdities. … Am I wrong or am I right?’

 

For a closer look at the notion of indefinable numbers, see the page:

 

Indefinable Numbers

Explore…

Why do people believe weird things? The Dead End Path of the Consciousness Myth True but unprovable? Gödel’s Proof: Step by Step Guide Alexander’s Horned Sphere The crackpot John Gabriel Turing’s “uncomputable” numbers The Chinese Room Gödel’s Proof: English Translation Numbers, chairs and unicorns On David Hilbert’s “On the Infinite” Cantor’s Grundlagen: English translation

Subject Categories

The Infinite Paradoxes Incompleteness Set Theory Logic & Language Platonism Philosophical Notes Math Curios Easy Footnotes Dark Mode

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Copyright   James R Meyer   2012 - 2025
https://jamesrmeyer.com