nLab second-order arithmetic

Idea

Second-order arithmetic is a theory of arithmetic dealing with natural numbers and, what makes it “second-order”, sets of natural numbers¹. SOA, or $Z_2$ as it is often denoted, is proof-theoretically weak in comparison with ZFC, but still strong enough to derive “almost all of undergraduate mathematics” (Friedman).

Definition

The language of SOA consists of two sorts, here denoted $N$ and $P N$, together with

• A constant $0$ of type $N$,

• An unary function symbol $s \colon N \to N$,

• Binary function symbols $+, \cdot \colon N \times N \to N$,

• A binary relation symbol $\lt$ of type $N \times N$.

• A binary relation symbol $\in$ of type $N \times P N$.

The axioms of SOA may be divided into two parts: the first part comprises the “first-order axioms” that deal only with the sort $N$, and defines which is known as Robinson arithmetic. The second part comprises induction and comprehension schemes that involve the symbol $\in$.

“First-order” axioms

Omitting the sort $P N$ and the symbol $\in$ from the language, the axioms in this section give Robinson arithmetic. The logic throughout is classical first-order (predicate) logic with equality.

For all the formulas in this section, it is tacitly understood that there are universal quantifiers at the heads of formulas, binding all variables which appear freely.

1. $s(m) = 0 \rightarrow \bot$.

2. $s(m) = s(n) \rightarrow m = n$.

3. $(0 = n) \vee (\exists_m s(m) = n)$.

4. $m + 0 = m$.

5. $m + s(n) = s(m+n)$.

6. $m \cdot 0 = 0$.

7. $m \cdot s(n) = (m\cdot n)+m]$.

8. $m \lt 0 \rightarrow \bot$.

9. $m \lt s(n) \leftrightarrow ((m \lt n) \vee (m=n))$.

10. $(0 = n) \vee (0 \lt n)$.

11. $((s(m) \lt n) \vee (s(m) = n)) \leftrightarrow m \lt n$.

“Second-order” axioms

Note: according to convention, lower-case letters refer to terms of type $N$, and upper-case letters to terms of type $P N$.

1. Comprehension scheme: for any formula $\varphi$ in the language of SOA with free variables $\vec{m} = m_1, \ldots, m_j$ and $\vec{X} = X_1, \ldots, X_k$,

   $$\forall \vec{m} \forall \vec{X} \exists Z \forall n (n\in Z \leftrightarrow \varphi(n))$$

   provided that the variable $Z$ does not appear in $\varphi$. (Using alpha-equivalence to rename variables, it's enough that $Z$ does not appear free in $\varphi$.)

2. Full induction scheme: for $\varphi$ any formula with a free variable $n$ and possible remaining free variables $\vec{m} = m_1, \ldots, m_j$ and $\vec{X} = X_1, \ldots, X_k$,

$$\forall \vec{m} \forall \vec{X} ((\varphi(0) \wedge \forall n (\varphi(n) \rightarrow \varphi(s(n))) \rightarrow \forall n \varphi(n))$$

The instance in the full induction scheme where $\varphi$ is the formula $n \in X$ is called simply the induction axiom. The induction axiom together with the comprehension scheme implies the full induction scheme.

Subsystems of SOA

The theory described above gives full second-order arithmetic. However, in reverse mathematics, one often studies subsystems of weaker proof-theoretic strength than SOA, by limiting in some way the comprehension scheme (often also beefing up the single induction axiom with more instances of the induction scheme, to offset the weakening). The main examples are given in Wikipedia; a standard reference is Simpson.

Many important subsystems for SOA have been the subject of an ordinal analysis.

Another important subsystem is intuitionistic second-order arithmetic, or second-order Heyting arithmetic. Here, we use intuitionistic logic rather than classical logic as the underlying first-order logic. In a subsystem without the induction axiom, we must add axioms stating that equality and order are decidable?:

• $(m = n) \vee ((m = n) \rightarrow \bot)$;

• $(m \lt n) \vee ((m \lt n) \rightarrow \bot)$.

Full (classical) SOA may be recovered from intuitionistic SOA by adding a similar axiom that set membership is decidable; in a subsystem without the comprehension scheme, this needs to be strengthened to an axiom scheme that every proposition (with any free variables) is true or false (effectively making the logic classical again).

Related concepts

• foundations

• elementary function arithmetic

• descriptive set theory

References

• Wikipedia Second order arithmetic

• Harvey Friedman’s Home Page

• Stephen G. Simpson, Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press (2009) (pdf)

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1. The logic that governs the language and theory of $Z_2$ is ordinary (first-order) predicate logic. The “second-order” aspect is really in the models, where one interprets the symbol $\in$ as membership in a background set theory, i.e., terms of type $P N$ are interpreted as subsets of the set that interprets the type $N$, and an extensionality axiom is in force. A full model is where $P N$ is interpreted as the full power set of $N$. When one sees absoluteness assertions such as “there is only one (full) model of SOA up to isomorphism,” it should be clear that this is meant with regard to a given background set theory. Cf. the fact that while there is, up to isomorphism, at most one natural numbers object $\mathbb{N}$ in a given topos, the set of global elements $\Gamma(\mathbb{N})$ might contain “non-standard elements” as viewed against the background $Set$. ↩