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 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