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.