Georg Cantor

1845–1918 · German mathematician, inventor of set theory

Topics: mathematics, learning

• There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.

  — Georg Cantor

• Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.

  — Georg Cantor

• The essence of mathematics lies entirely in its freedom.

  — Georg Cantor

• Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.

  — Georg Cantor

• The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.

  — Georg Cantor

• In mathematics the art of asking questions is more valuable than solving problems.

  — Georg Cantor

• The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set.

  — Georg Cantor