In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative...

second-order arithmetic. Just as in first-order logic, second-order logic may include non-logical symbols in a particular second-order language. These...

of arithmetic in second order logic that is called second order arithmetic. It has only one model, unlike the corresponding theory in first-order logic...

the second-order and first-order formulations, as discussed in the section § Peano arithmetic as first-order theory below. If we use the second-order induction...

Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous...

includes quadratic terms Second-order arithmetic, an axiomatization allowing quantification of sets of numbers Second-order differential equation, a differential...

In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers. This is the theory associated...

theorem, is undecidable in (first-order) Peano arithmetic, but can be proved in the stronger system of second-order arithmetic. Kirby and Paris later showed...

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej...

stated that the axioms he considered for arithmetic were the ones given in Hilbert (1900), which include a second order completeness axiom. In the 1930s, Kurt...

second-order arithmetic in the sense that each of the statements can be derived from each other in the weak system of primitive recursive arithmetic (PRA)...

∀x(Fx ↔ Gx). V* is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic. Basic Law V can simply be replaced...

Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929...

an analogue of the axiom of constructibility for subsystems of second-order arithmetic. A few results stand out in the study of such analogues: John Addison's...

of second-order arithmetic and reverse mathematics. The field of proof theory includes the study of second-order arithmetic and Peano arithmetic, as...

the syntax of formal logic within first-order arithmetic. Each expression of the formal language of arithmetic is assigned a distinct number. This procedure...

extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have...

a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion). In 2004, the result was...

that HP and suitable definitions of arithmetical notions entail all axioms of what we now call second-order arithmetic. This result is known as Frege's theorem...

Peano Arithmetic using transfinite induction up to ordinal ε0. Ordinal analysis has been extended to many fragments of first and second order arithmetic and...

first-order logic. A theory about a topic, such as set theory, a theory for groups, or a formal theory of arithmetic, is usually a first-order logic together...

transfinite induction of arithmetical statements for R {\displaystyle R} . Some theories, such as subsystems of second-order arithmetic, have no conceptualization...

are still fairly significant (in ascending order): The proof-theoretic ordinal of second-order arithmetic. A possible limit of Taranovsky's C ordinal...

suitable vertex. In this case, Kőnig's lemma is provable in second-order arithmetic with arithmetical comprehension, and, a fortiori, in ZF set theory (without...

that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic or Zermelo-Fraenkel set theory)...

In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by Raphael M. Robinson in 1950...

elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary...

Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory...

be proven in much weaker systems than ZFC, such as Peano arithmetic and second-order arithmetic (as explored by the program of reverse mathematics). Saunders...

1939, it presents fundamental mathematical ideas and introduced second-order arithmetic. 1934/1939 (Vol. I, II) First German edition, Springer 1944 Reprint...