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II Inconsistence of the transfinite number ω and the set ℕ

April 2023. For more than 2 millennia, Aristotle's "infinitum actu non datur", "there is no limited infinity", dominated the perception of the infinite. End of the 19th century Georg Cantor introduced the actual infinity or transfinite with the postulate of limits, even levels in infinity, with his set theory. The result was a foundational crisis of mathematics. Cantor's doctrine was not axiomatically founded. The axiomatic system of Zermelo and Fraenkel then formally legitimized the transfinite. The crisis is considered to be overcome thereby. Cantor's transfinite was initially highly controversial, but ultimately prevailed. Today, in its axiomatic form, it constitutes one of the foundations of mathematics. In the following it will be shown that both, the axiomatic and Cantor's justification of the transfinite number ω and the set ℕ