Burali-Forti paradox

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Cesare Burali-Forti, who in 1897 published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor. Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book "Principles of Mathematics", he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name.

We will prove this by reductio ad absurdum.

We've deduced two contradictory propositions (${\displaystyle \Omega <\Omega }$ and ${\displaystyle \lnot \Omega <\Omega }$) from the sethood of ${\displaystyle \Omega }$ and, therefore, disproved that ${\displaystyle \Omega }$ is a set.

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