Logic and Language

Copyright   James R Meyer    2012 - 2023 https://www.jamesrmeyer.com

# Hilbert’s Tenth Problem

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?

## 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 Axiom of Choice

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

## The Origins of Transfinite Numbers

Georg Cantor How and why did Cantor introduce the notion that there are different levels of infinity?

Most accounts gloss over various aspects of Cantor’s writings that can appear to us today as strange, if not manifestly incorrect.

Here we give a logical appraisal of Cantor’s methodology and thought processes that are indicated by his writings, see:

The Origins of Transfinite Numbers

## 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

## Indefinable Numbers

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

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Copyright   James R Meyer   2012 - 2023
https://www.jamesrmeyer.com