Logic and Language
Load the menuLoad the menu


Copyright   James R Meyer    2012 - 2024 https://www.jamesrmeyer.com

BANNER CONTENT

The Continuum Hypothesis

The Continuum Hypothesis is the assertion:

“There is no set whose limitlessness of size is more than the limitlessness of the quantity of natural numbers but less than the limitlessness of the quantity of real numbers.”

 

Whether you think that is in any way meaningful or not is up to you. Most mathematicians hold that the Continuum Hypothesis cannot be proved in conventional set theories, and that it cannot be disproved in conventional set theories - they accept that Paul Cohen showed that conventional set theories cannot prove it (Footnote: Paul J Cohen: The Independence of the Continuum Hypothesis, Proceedings of the National Academy of Sciences of the United States of America 50 (6): 1143-1148 (1963).) and that Kurt Gödel showed that conventional set theories cannot disprove it. (Footnote: Kurt Gödel: The Consistency of the Continuum Hypothesis, Princeton University Press, 1940.)

 

Accordingly, the definitions of conventional set theories do not provide sufficient means to either prove the Continuum Hypothesis or disprove it.

 

But even if mere humans might never be deduce the answer, Platonists believe that there is a ‘true’ answer to the question, so that if you are a Platonist, you will believe that the Continuum Hypothesis is either ‘true’ or ‘false’. And if you accept Fantasy-set theory, you can either:

  • believe that the Continuum Hypothesis is ‘true’, or
  • believe that the Continuum Hypothesis is ‘false’.

and, regardless of which you choose, no set theorist can ever prove you wrong.

 

And if you accept conventional set theory, and if you believe that the Continuum Hypothesis is ‘false’, you could simply assume that there are say, four levels of limitlessness between the limitlessness of natural numbers and the limitlessness of real numbers; and no conventional set theorist can ever prove you wrong.

 

Or you could assume that there are no levels of limitlessness between the limitlessness of natural numbers and the limitlessness of real numbers; and no conventional set theorist can ever prove you wrong.

 

Indeed, there is no limit to the number of persons who could hold different assumptions about the number of levels of limitlessness between the natural numbers and the real numbers; and no conventional set theorist can ever prove any of them wrong. As one observer remarked:

 

“I suspect that it is futile to wonder now what Gödel imagined to be the ‘real’ cardinal of the continuum. Those earnest specialists who still search for that cardinal may call to mind that infamous image of the philosopher – a blind man in a dark cellar looking for a black cat that is not there” (Footnote: Saunders Mac Lane, in the essay “Is Mathias an Ontologist?” in the book Set Theory of the Continuum (Judah, Just, and Woodin, eds) Springer-Verlag, 2012.)

 

Conventional set theorists don’t like it to be expressed in this way; in ordinary English it just sounds so inanely absurd. And so they simply try to hide the inanity of it all by euphemisms which sound rather grand, such as:

 

“The Continuum Hypothesis is independent of the axioms of set theory.”

 

Georg Cantor was the man who was the originator of the theory of different sizes of infinity that led to the problem of the Continuum Hypothesis. Initially he was quite confident that it was correct, and also that he would soon solve it (obviously he didn’t). You can read online an English translation of one of Cantor’s major works, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a general theory of sets), which lays out his philosophy on different sizes of infinity.

 

For an overview of today’s set theories, see the pages that give an overview of set theory, starting at Overview of set theory: Part 1: Different types of set theories.

Footnotes:

Interested in supporting this site?

You can help by sharing the site with others. You can also donate at Go Get Funding: Logic and Language where there are full details.

 

 

As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Frivolous, irrelevant comments.
Comments devoid of logical basis.
Derogatory comments.
Long-winded comments.
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical terms are correctly formatted.


Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous user names for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it incorrectly and rewrite it again - still erroneously, or else attack something else on my site - erroneously. After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.


Difficulties in understanding the site content are usually best addressed by contacting me by e-mail.

 

Based on HashOver Comment System by Jacob Barkdull

Copyright   James R Meyer   2012 - 2024
https://www.jamesrmeyer.com