Undecidability of Gödel's first incompleteness theorem. Gödel and Hilbert mathematics

1 Васил Пенчев Неразрешимост на т. нар. първа теорема на Гьодел за непълнотата. Гьоделова и Хилбертова математика Abstract: Can the so-called first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected... More

1 Васил Пенчев Неразрешимост на т. нар. първа теорема на Гьодел за непълнотата. Гьоделова и Хилбертова математика Abstract: Can the so-called first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected with some kind of self-reference. Gödel built his proof on the ground of self-reference: a statement which claims its unprovability. So, he demonstrated that undecidable propositions exist in any enough rich axiomatics (i. e. such one which contains Peano arithmetic in some sense). What about the decidability of the very first incompleteness theorem? We can display that it fulfills its conditions. That’s why it can be applied to itself, proving that it is an undecidable statement. It seems to be a too strange kind of proposition: its validity implies its undecidability. If the validity of a statement implies its untruth, then it is either untruth (reductio ad absurdum) or an antinomy (if also its negation implies its validity) Less