I saw in an answer to a recent question (www.ikhebeenvraag.be/ question/35736) that the “number” of real numbers is aleph -1. But has that been proven? I thought this was just an unprovable statement that you may or may not add (or its negation) to the math. How does that work now?
Answer
Dear Nicole
That’s right. In the current system of axioms for set theory (“ZFC”) this is an undecidable statement. This statement is called the “continuum hypothesis” and it is independent of ZFC: so you can indeed add both the continuum hypothesis and its negation to the axiomatic system.
The cardinality of the natural numbers (and by extension: of all infinite, countable sets) is given by ℵ0 (“alef-0”) noted. The next strictly larger cardinality is denoted with ℵ1. For the real numbers, it can be shown that the cardinality is strictly greater than that of the natural numbers (the set of the real numbers is said to be “uncountable”).
Since there is a byejection between the real numbers and all subsets of natural numbers (this is the so-called “power set” of the natural numbers), the cardinality of the real numbers is exactly 2^(ℵ0).
With these agreements, the continuum hypothesis comes down to the equality 2^(ℵ0) =1 and can be expressed in words as “there exists no set with a cardinality strictly between that of the natural (ℵ0) and the real (2^(ℵ0)) numbers”.
regards
Tom
Answered by
Tom Dorissen
Avenue des Pélain 2 1050 Ixelles
http://www.vub.ac.be/
.