In set theory and other branches of mathematics, **ב**_{2} (pronounced *beth two*), or **2**^{c} (pronounced *two to the power of* c), is a certain cardinal number. It is the 2nd beth number, and is the result of cardinal exponentiation when 2 is raised to the power of *c*, the cardinality of the continuum. This number 2^{c} is the cardinality of many sets, including: - The power set of the set of real numbers, so it is the number of subsets of the real line, or the number of sets of real numbers;
- The power set of the power set of the set of natural numbers, so it is the number of sets of sets of natural numbers;
- The set of all functions from the real line to itself;
- The power set of the set of all functions from the set of natural numbers to itself, so it is the number of sets of sequences of natural numbers;
- The set of all real-valued functions of
*n* real variables to the real numbers. Some early set theorists hypothesised the equation stating that 2^{c} is equal to the 2nd aleph number. It turns out that the truth of this equation (*) cannot be determined from the standard Zermelo-Fraenkel axioms of set theory; it is true in some models and false in others. (*) is a part of the generalized continuum hypothesis (GCH), but it is possible that (*) is true while the full GCH is false. On the other hand, if (*) is true, then the ordinary continuum hypothesis (CH) must follow, but again it is possible that CH is true while (*) is false. *This article or a past revision is based on the Mandelbrot Set Glossary and Encyclopedia (**http://www.mrob.com/pub/muency.html*), copyright © 1987_2003 Robert P. Munafo, which is made available under the terms of the GNU Free Documentation License. |