Unlike a weaker version invented by Pressburger that omits(*) multiplication, this system is not decidable.
Laurie Kirby and Jeff Paris showed that it is not possible to prove that every Goodstein sequence terminates at zero in this system; Paris and Leo Harrington used model theory and the condition that the cardinality of x is greater than or equal to its minimum to show that a strengthened finite Ramsey theorem is true in this system, but is similarly unprovable. Unlike a weaker version invented by Pressburger that omits(*) multiplication, this system is not decidable. A proof that defined a quantity epsilon sub zero using transfinite induction was used to attempt to answer Hilbert's second problem; that proof by Gerhard Gentzen demonstrated this system's consistency. One statement in this system defines a "successor" function that is used in definitions of induction, others give the properties of the equality relation, and the first of these define zero as part of the set. For 10 points, name this set of axioms defining the natural numbers, invented by an Italian.
Laurie Kirby and Jeff Paris showed that it is not possible to prove that every Goodstein sequence terminates at zero in this system; Paris and Leo Harrington used model theory and the condition that the cardinality of x is greater than or equal to its minimum to show that a strengthened finite Ramsey theorem is true in this system, but is similarly unprovable.
Peano axioms or postulates or arithmetic [accept Dedekind-Peano axioms/postulates or first-order arithmetic]