Controversy over Cantor's theory

In <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a>, the theory of <a href="/facts/Infinite_set/9rTN0xGH">infinite sets</a> was first developed by <a href="/facts/Georg_Cantor/k5v1DvW4">Georg Cantor</a>. Although this work has become a thoroughly standard fixture of classical <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, it has been criticized in several areas by mathematicians and philosophers.
<a href="/facts/Cantor%2527s_theorem/kGNIBpjL">Cantor's theorem</a> implies that there are sets having <a href="/facts/Cardinality/m6VPojwT">cardinality</a> greater than the infinite cardinality of the set of <a href="/facts/Natural_numbers/u65og8JE">natural numbers</a>. Cantor's argument for this theorem is presented with one small change. This argument can be improved by using a definition he gave later. The resulting argument uses only five axioms of set theory.
Cantor's set theory was controversial at the start, but later became largely accepted. Most modern mathematics textbooks implicitly use Cantor's views on <a href="/facts/Mathematical_infinity/cF506uD7">mathematical infinity</a>. For example, a <a href="/facts/Line_(geometry)/gJqsr4Gj">line</a> is generally presented as the infinite set of its points, and it is commonly taught that there are more real numbers than rational numbers (see <a href="/facts/Cardinality_of_the_continuum/ob578Q9l">cardinality of the continuum</a>).

Controversy over Cantor's theory open-in-new

Controversy over Cantor's theory