Second-order arithmetic

Second-order arithmetic or Z2 is a theory of arithmetic allowing quantification over the natural numbers as well as sets of natural numbers.

Language
The language of second-order arithmetic allows us to discuss two types of objects, namely number themselves, or sets of numbers. It is an extension of predicate calculus with the following:

The language of second-order arithmetic is expressive enough to allow us to talk about not only natural numbers and subsets of these, but also about integers, rational numbers and real numbers, and even countable sets (i.e. sequences) of real numbers.
 * Infinitely many numeric variables \(x_0\), \(x_1\), ... and infinitely many set variables \(X_0\), \(X_1\), ...
 * Constant symbol \(0\), called zero
 * Three binary relation symbols \(=,<,\in\), called equality, comparison and set membership. The first two are relations between numbers, and the last one is a relation between a number and a set
 * Unary function symbol \(S(x)\), called successor, mapping numbers to numbers
 * Two binary function symbols, \(+(a,b),\cdot(a,b)\), called addition and multiplication respectively, which map pairs of numbers to a single number. They are often denoted by \(a+b,a\cdot b\).

Axioms
The theory of second-order arithmetic consists of the axioms of Peano arithmetic, plus:


 * \(\forall X (0 \in X \wedge \forall x (x \in X \Rightarrow Sx \in X) \Rightarrow \forall x : x \in X)\), the second-order induction axiom. It states that if a set contains 0 and contains the successor of each of its own members, then that set contains all natural numbers.
 * For every formula \(\phi\) where \(X\) is not a free variable: \(\exists X \forall x (x \in X \Leftrightarrow \phi(x))\), the axiom schema of comprehension. It tells us that every predicate over the natural numbers defines a subset of the natural numbers (namely the set of these number which satisfy \(\phi\)).

Subsystems
Z2 is an extremely strong formal theory, and in many cases we do not require all of its strength to prove a desired result. We naturally ask what happens when we weaken Z2. Harvey Friedman started a project, now known as, which addresses questions about what axioms are necessary to prove a given theorem, and what axioms are not strong enough to prove it.

Reverse mathematics focuses on subsystems of second-order arithmetic, which have been extensively studied to measure strength of their theorems. Many "everyday" theorems in mathematics have strengths corresponding to one of the following subsystems, often called the "big five".

RCA0
RCA0, for "recursive comprehension axiom," is the weakest of the Big Five. It consists of:


 * First-order schema of induction limited to \(\Sigma_1^0\) formulas (see )
 * Axiom schema of comprehension, limited to \(\Delta_1^0\) formulas (see )
 * Axiom schema of comprehension, limited to \(\Delta_1^0\) formulas (see )

This is the so called base system, and all other subsystems have RCA0's axioms. RCA0 is a weak system, but it can prove many basic properties of natural numbers and real numbers, including the uniqueness of a limit of a sequence and the ). RCA0 can prove most "everyday" properties of natural numbers and real numbers.

RCA0 only has enough strength to prove the existence of objects which are recursive. For example, the set existence axioms of this theory lack the strength to prove the existence of uncomputable functions.

WKL0
WKL0 stands for "weak König's lemma." We form WKL0 by extending RCA0 with a weak form of :


 * Let \(2^{<\omega}\) be the tree of all finite binary sequences, and let \(T\) be one of its finite subtrees. Then \(T\) has an infinite branch.

RCA0 canot prove this lemma — recalling that RCA0 cannot prove the existence of uncomputable sets, we can create an infinite computable tree \(T\) which has no computable infinite branches. Consequently, WKL0 is a proper extension of RCA0.

It can be shown that many natural statements are equivalent (over base theory RCA0) to WKL0, e.g. Heine-Borel theorem for closed intervals, or Riemann-integrability of continuous functions.

ACA0
Arithmetical comprehension axiom. We get this by adjoining to RCA0 axiom schema of comprehension for all arithmetical formulas (ones contained in \(\Sigma_n^0\) for some \(n\in\Bbb N\)). It can be shown that formulas provable in ACA0 which can be expressed in first-order arithmetic are precisely the formulas provable by Peano arithmetic (one says that ACA0 is conservative over Peano arithmetic for arithmetical sentences). Like before, it can be shown that ACA0 is a proper extension of WKL0, although the proof in this case is not as simple.

ACA0 is strong enough to prove all the statements required to work well with real (or complex) analysis, including or full König's lemma (for trees on finite sequences of natural numbers) (in fact, ACA0 is provably equivalent to these two statements).

ATR0
This one stands for "arithmetical transfinite recursion". This system is ACA0 together with axiom schema stating, intuitively, that if we have countable well-ordering (coded in a certain way) and so called arithmetical functional, we can iterate application of this functional transfinitely by following the mentioned well-ordering. This means that certain set constructions can be accomplished by iterating a simple construction transfinite number of times.

Addition of this axiom schema makes it possible to build a theory of ordinal numbers in second-order arithmetic. An example of strength of this schema is that it's sufficient to prove the consistency of ACA0, which means that it is significantly stronger than ACA0 (see ).

Important single theorem which is equivalent (over RCA0) to ATR0 is that every two countable well-orderings are comparable, so that ordinal numbers form a linear order.

\(\Pi_1^1-CA_0\)
The last, and the strongest system in the big five is RCA0 extended by \(\Pi_1^1\) comprehension axiom schema, hence the abbreviation. This is so called impredicative system, because it's axioms, in a way, assert existence of objects which depend on the object defined (this is because defining set \(X\) using \(\Pi_1^1\) formula involves statement which is supposed to hold for all sets, including \(X\) itself).

It's clear that \(\Pi_1^1-CA_0\) contains all the axioms of ACA0, and it can be shown that it also proves all instances of ATR0 axioms, which shows that \(\Pi_1^1-CA_0\) can build basic theory of ordinal numbers. However, as one can show, statement that given order is well-order is a \(\Pi_1^1\) statement, which means that this last system can show existence of sets definitions of which explicitly use notions of well-ordering, which, in big part, is where the strength of the system comes from.