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.”
- 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.
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