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

AI systems agree: There is a fundamental error in Gödel’s Incompleteness Proof

 


Kurt Gödel
Kurt Gödel

AI systems have definitively affirmed that there is a fundamental flaw in Gödel’s Proof of Incompleteness.

 

In 1931 the mathematician Kurt Gödel claimed to have proved that in every possible formal mathematical system is incomplete - that there are statements within that system but which can never be proved to be true in that system, but which are nevertheless true.

 

But how rigorous is Gödel’s Proof ? Did he make unfounded assumptions which render his proof illogical and invalid?

 

Current AI systems certainly think so. They agree that, in fact, there is a fundamental flaw in Gödel’s proof.

 

 

For more on this see:

AI on Gödel’s Incompleteness Proof

Goodstein Sequences

 


Reuben Goodstein
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

Chaitin’s “Unknowable” Number

 


Gregory Chaitin
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

The Axiom of Choice

 


Ernst Zermelo
Ernst Zermelo

 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

 


David Hilbert
David Hilbert

 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

 


Henri Lebesgue
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

Indefinable Numbers

 


Julius König
Julius König

 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 Chaitin’s Unknowable Number 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