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(11) Can there exist a set S whose cardinality |S is the largest pos- sible cardinality? We have seen that the cardinality of any set S is strictly smaller than the cardinality of its power set P(S) Can you use this to define a sequence of countably infini many cardinalities? For example, start with the set of natural num- bers N. Its cardinality is No. Now take the power set P(N) of N and then the power set P(P(N)) of the power set of N and so on. Define the cardinalities 2o NoIN (say beth zero, -is the second letter in the Hebrew alphabet) and inductively P(N) P(P(N))| and so on. Discuss the sequence of cardinalities Io, I1,2, . What is the relation between Dk+ and There is also another sequence No,¡א,א2, of cardinalities, where Ni is the cardinality of the set of all ordinal numbers. But we will not discuss this sequence and concepts here as this would need a thorough introduction into axiomatic set theory Let us just make a final remark here. For example, one could ask if there exists a set S with cardinality ISI which is between the cardinality of the natural numbers and the cardinality of the real nurnbers c, i, e. א 0< Isl < c? In other words, is the next cardinality after ) the cardinality of the continuum c2o and therefore N2? This is the content of the famous Continuum Hypothesis con- jectured by Cantor in 1877, but shown to be independent (by Gödel (1940) and P. J. Cohen (1963)) of the basic (Zermelo- Fraenkel) axioms of set theory. So similar as with the paral lel postulate in Euclidean geometry, which gave rise to Nor Euclidean geometry, there exist axiom systems, where the Con tinuum Hypothesis is true or not giving rise to different foun- dations of set theory and therefore of mathematics
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