It seems to be common knowledge that $\mathsf{ACA}_0$ can be finitely axiomatized, see for example

Can a finite axiomatization of PA be expressed in a finitely axiomatizable first order set theory?

However, I cannot seem to find a reference where $\mathsf{ACA}_0$ is given explicitly with the finite list of axioms. It is always defined as an extension of $\mathsf{PA}$ which already has an infinite axiom scheme.

Is there some reference giving the finite axiom list explicitly or is the above result of finite axiomatizability a non-constructive existential statement?

I hope this is the right place to ask for references. Delete this and PM/Mail me if not.


2 Answers 2


A finite axiomatisation for $\mathsf{ACA}_0$ is given by lemma VIII.1.5 of Simpson [2009], on pages 311–312. Fix a universal lightface $\Pi^0_1$ formula $\pi(e, m_1, X_1)$ with exactly the displayed free variables. Then we can give a finite axiomatisation of $\mathsf{ACA}_0$ as follows.

1) The basic axioms, essentially the axioms of Peano arithmetic minus the induction scheme (see definition I.2.4 on page 4 of Simpson [2009]).

2) A pairing axiom,

$$\forall{X}\forall{Y}\exists{Z} (Z = X \oplus Y).$$

3) $\Sigma^0_1$ induction in the form $$\forall{X}(\neg\pi(e, 0, X) \wedge \forall{m}(\neg\pi(e, m, X) \rightarrow \neg\pi(e, m + 1, X))) \rightarrow \forall{m} \neg\pi(e, m, X)).$$

4) $\Sigma^0_1$ comprehension in the form

$$\forall{X}\exists{Y}\forall{m}(m \in Y \leftrightarrow \neg\pi(e, m, X)).$$

The universal $\Pi^0_1$ formula $\pi$ will of course be something of the sort discussed in Peter Smith's blog post.

S. G. Simpson. Subsystems of Second Order Arithmetic. Cambridge University Press, 2nd edition, 2009. doi:10.1017/cbo9780511581007.


I have a blogpost about this at http://www.logicmatters.net/2007/09/14/aca0-3-finite-axiomatizability/ (as has been pointed out!).

The finite axiomatization you get this way, however, is not particularly pretty or illuminating or, you might think, so "natural".

  • $\begingroup$ I have not parsed the article yet, just noticed that it has no list of formulas. I do not care for the prettyness/minimal-ness of the formulas. Has anybody explicitly enumerated these as first order formulas such that I could plug them into a FOL proof searcher to let it show the basic pa facts such as S0 != 0? Will there be more like some $\approx 10^n$ or more like $> 10^{10^n}$ axioms? $\endgroup$
    – masterxilo
    Oct 2, 2014 at 15:14
  • 1
    $\begingroup$ It may be worth pointing out that the formula you give in the blog post only has one set parameter, so you need an additional pairing axiom that says for every $X, Y \subseteq \mathbb{N}$, $X \oplus Y$ exists. Alternatively, just have two set parameters in the comprehension axiom. $\endgroup$ Oct 2, 2014 at 15:32
  • $\begingroup$ @masterxilo: see Benedict Eastaugh's answer for that, and my explanation for the motivation of those axioms. $\endgroup$ Oct 2, 2014 at 23:42

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