1. Russell's paradox: a paradox about set theory put forward by British philosopher Bertrand Russell. Simply put, Russell's paradox points out that if all sets can be described as an element of themselves, can this set also be described as an element of itself? This leads to the questioning of some basic assumptions of set theory.
2. The theory of infinity and real numbers put forward by Georg Cantor, a German mathematician. Cantor's paradox reveals that the potential of real number set is greater than that of natural number set, which leads to the in-depth discussion of infinity and infinitesimal.
3. Hilbert paradox: a geometric problem put forward by German mathematician david hilbert. Hilbert's paradox involves the axiom of parallelism in Euclidean geometry, that is, "there is only one straight line parallel to the known straight line through a point outside the straight line". Hilbert tried to prove that this axiom was independent of other axioms, but in the end he failed.
4. Blary-Forti Paradox: The paradox about the properties of real number set put forward by Italian mathematicians Alberto Blary and Francisco Forti. Blary-Foday paradox reveals that the potential of a real number set is less than that of its power set, which leads to a re-examination of the properties of real number sets.
5. Cantor-Bernstein Theorem: A theorem on the properties of real number sets proposed by German mathematicians Georg Cantor and Felix Bernstein. Cantor-Bernstein theorem reveals that the potential of a real number set is greater than the sum of the potentials of all its subsets, thus further deepening the understanding of the properties of real number sets.