After having worked, instructed by Hermann von Helmholtz, on electrical stimulation of nerves, he switched to mathematics.

As a post-doc he completed his mathematical studies in Berlin attending lessons by Leopold Kronecker and Karl Weierstraß.

His work on polynomial ideals, discriminants and elimination theory can be considered as a link between Leopold Kronecker and David Hilbert as well as Emmy Noether.

The foundations of set theory are a formalization and legalization of facts which are taken from the internal view of our consciousness, such that our "scientific thinking" itself is an object of scientific thinking.But mainly he is remembered for his contributions to and his opposition against set theory.

Kőnig found a simple method involving decimal numbers which had escaped Cantor.

In 1904, at the third International Congress of Mathematicians at Heidelberg, Kőnig gave a talk to disprove Cantor's continuum hypothesis.

Ernst Zermelo, the later editor of Cantor's collected works, found the error already the next day.

In 1905 there appeared short notes by Bernstein, correcting his theorem, and Kőnig, withdrawing his claim.

Contrary to Cantor, presently the majority of mathematicians considers undefinable numbers not as absurdities.

This assumption leads, according to Kőnig, in a strangely simple way to the result that the continuum cannot get well-ordered.

The last part of his life Kőnig spent working on his own approach to set theory, logic and arithmetic, which was published in 1914, one year after his death.

In a letter to Philip Jourdain in 1905 he wrote: You certainly heard that Mr. Julius Kőnig of Budapest was led astray, by a theorem of Mr. Bernstein which in general is wrong, to give a talk at Heidelberg, on the international congress of mathematicians, opposing my theorem according to which every set, i.e., every consistent multitude can be assigned an aleph.