Analog of Gödel's incompleteness theorem Using a Falsifiability Predicate

K. Gödel's incompleteness theorems are considered in relation to the formal Dedekind–Peano arithmetic. A falsifiability predicate is used to construct a formula that formally expresses its own nonfalsifiability. And unsolvability of that formula is proved. This proves the incompleteness of formal arithmetic, that is, the conclusion of the first theorem is confirmed. At the same time, with this representation of (un)provability, the main conclusion of the second theorem turns out to be false. It follows that the second theorem is independent of the first, which denies the generally accepted statement about the inseparable connection between Gödel's first and second incompleteness theorems.