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:
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:
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:
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:
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:
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:
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Did you hear about the hungry clock?
It goes back four seconds.
I have set up this website to allow a user to switch to a dark mode, but which also allows the user to revert back to the browser/system setting. The details of how to implement this on a website are given at How to setup Dark mode on a web-site.
The page Understanding sets of decreasing intervals explains why certain definitions of sets of decreasing intervals are inherently contradictory unless limiting conditions are included, and the page Understanding Limits and Infinity explains how the correct application of limiting conditions can eliminate such contradictions. The paper PDF On Smith-Volterra-Cantor sets and their measure has additional material which gives a more formal version.
How to set up a system for easy insertion or changing of footnotes in a webpage, see Easy Footnotes for Web Pages.
After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details the confusion of levels of language. The paper can be seen at The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.
To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers published between 1879 and 1884 under the overall title of Über unendliche lineare Punktmannig-faltigkeiten, which were published between 1879 and 1884. I now have English translations of Part 1, Part 2, Part 3 and the major part, Part 5 (Grundlagen). There is also a new English translation of Cantor’s “A Contribution to the Theory of Sets”.
A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for strict rigor.
The pages of this website are set up to give a good printed copy without extraneous material.
How to set up a system for easy insertion or changing of footnotes in a webpage, see Easy Footnotes for Web Pages.
After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details the confusion of levels of language. The paper can be seen at The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.
To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers published between 1879 and 1884 under the overall title of Über unendliche lineare Punktmannig-faltigkeiten, which were published between 1879 and 1884. I now have English translations of Part 1, Part 2, Part 3 and the major part, Part 5 (Grundlagen). There is also a new English translation of Cantor’s “A Contribution to the Theory of Sets”.
A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for strict rigor.
The pages of this website are set up to give a good printed copy without extraneous material.
Copyright James R Meyer 2012 - 2024
https://www.jamesrmeyer.com
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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