Continuum problem

Japanese: 連続体問題 - れんぞくたいもんだい

The cardinality of all real numbers is represented by c or א, and is called the cardinality of the continuum or the cardinality of the continuum. G. Cantor used the diagonal argument to prove that the cardinality of the continuum is truly greater than the countable cardinality (the cardinality of all natural numbers) a. He further conjectured that there is no set with cardinality b such that a < b < c, and attempted to prove this but was unsuccessful. The hypothesis that the next step after countable cards is the cardinality of the continuum is called the continuum hypothesis. After Cantor, axiomatic set theory was developed, which reconstructs Cantor's naive set theory using axioms, and research continued into the validity of the continuum hypothesis.

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

実数全体の濃度をcまたはאで表し，連続の濃度もしくは連続体の濃度という。G.カントルは，対角線論法によって連続体の濃度は可算の濃度(自然数全体の濃度)aより真に大きいことを示した。さらに彼はa＜b＜cであるような濃度bをもつ集合は存在しないと予想し証明を試みたが成功しなかった。可算の濃度の次は連続体の濃度であるという仮説を連続体仮説という。カントル以降カントルの素朴な集合論を公理を使って再構成する公理的集合論が展開され，連続体仮説の正否をめぐって研究が続けられた。

出典　株式会社平凡社世界大百科事典第２版について　情報

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