On Equivalents of Well-foundedness An experiment in Mizar

Four statements equivalent to well-foundedness (well-founded induction, existence of recursively deened functions, uniqueness of recursively deened functions , and absence of descending !-chains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively deened functions implies well-foundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project.