The first published paradox is Blary's Forty Paradox, which means that ordinal numbers form an ordered set according to their natural order. By definition, this well-ordered set also has an ordinal ω, which should belong to this well-ordered set by definition. However, according to the definition of ordinal number, the ordinal number of any segment in ordinal number sequence is greater than any ordinal number in that segment, so ω should be greater than any ordinal number, so it does not belong to ω. This was put forward by Blary Forti in an article read at the Balomo Mathematics Conference on March 28th, 1997. This is the first published modern paradox, which aroused the interest of the mathematical community and led to heated discussions for many years. There are dozens of articles discussing paradox, which greatly promotes the re-examination of the basis of set theory.

Blary Foday himself thinks that this contradiction proves that the natural order of this ordinal number is only a partial order, which contradicts the result ordinal number set proved by Cantor a few months ago. Later, Blary Foday didn't do this work either.

In his Principles of Mathematics, Russell thinks that although the ordinal set is fully ordered, it is not well ordered, but this statement is unreliable because the first paragraph of any given ordinal number is well ordered. French logician Jourdain found a way out. He distinguished between compatible sets and incompatible sets. This distinction has actually been used privately by Cantor for many years. Soon after, Russell questioned the existence of ordinal set in an article in 1905, and Zemelo also had the same idea, and later many people in this field held the same idea.