Journey Through Genius: The Great Theorems of Mathematics Chapter Summary & Analysis - Cantor and the Transfinite Realm Summary

Cantor and the Transfinite Realm Summary and Analysis

Dunham continues to focus on Cantor and his exploration of the infinite. Cantor develops a way to compare the relative sizes of his cardinal numbers which represent infinite sets, and his efforts are proven by other mathematicians. Cantor has defined two types of transfinite cardinals, אₒ and c.

Cantor suspects there are other transfinite cardinals even greater than c, and he sets out to find them. He first tries extending the continuum between 0 and 1 into two dimensions, but finds that the set is still equal to c.

Cantor is finally successful in proving that there are other transfinite cardinals greater than c and his theorem that does so is the great theorem Dunham chooses as the central one of the final chapter. Cantor makes his proof by refining and expanding set theory, which allows for an ever increasing number of sets of sets, and more sets of even...
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