# The Undervalued Kleene Hierarchy

TLDR;

(i) A general introduction to the most underrated tool of proof theory: the *Kleene–Mostowski Hierarchy,* cf. [Arithmetical hierarchy (wikipedia)](https://en.wikipedia.org/wiki/Arithmetical_hierarchy). (ii) The claim is that this diagnostic tool should not remain confined to a specialty; it should be treated as standard equipment wherever one reasons about agents, verification, security and uniform procedures. (iii) Real life examples from chess and a corporate toy model.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/f9624720fad85b859e3e324642f8af6fe2c8590cd71cff2f.png)

If the language of Nature is math, the grammar is recursion.

1.
==

Observation. One thing to love about proof theory is its implicated pessimism: start as if the cause is already lost, set the task of *failing elegantly*.

Surprisingly, major failures of formal closure*,* often simultaneously opened a new methodological frontiers. It seems that whenever a logical framework reaches a boundary it cannot constructively cross, the mechanism that exposes the boundary tends to become constructive in a higher sense.

List. (i) *Diagonalization* begins as an uncountability device, becomes the standard method of inspection; (ii) *Gödel Numbering* supplies the *first compiler:* arithmetization of provability; (iii) *Halting* defines the design space of computation, (iv) while $\lambda$-calculus becomes th prototype for programming languages. (v) *Kolmogorov complexity* turns a methodological slogan into a mathematical object.

Analogy. One runs into a wall because forward motion is no longer available. They climb. Soon the wall is no longer an obstruction, it becomes indispensable. 

2.
==

Proposition. Among the boundary-tools that *should* be more widely used, the *Kleene–Mostowski Hierarchy* is a strong candidate. Its universality makes it useful in all contexts involving a problem of *any* kind.

$$
\{\Delta_n,\Sigma_n, \Pi_n \}
$$

Characterization. Standard in computability, proof theory, and reverse mathematics. Often replaced by coarser proxies, “ad hoc notions”, “$\mathcal{N\!P}$ hard”. Those proxies are useful, but they conceal key distinctions.

Consider,

$$
\begin{array}{|c|c|l|} \hline \text{Level} & \text{Prenex Form} & \text{Example / Reading} \ \hline \Delta_0 & \forall u < t,\exists v < s,\varphi(u,v) & 5 + n = 7\;(\text{bounded search}) \ \hline \Sigma_1 & \exists x,\varphi(x) & \exists d,(d\mid 91 \wedge 1<d<91)\;(\text{semi-decidable}) \ \hline \Pi_1 & \forall x,\varphi(x) & \forall n,(n>0 \rightarrow n+1>n)\;(\text{universal}) \ \hline \Sigma_2 & \exists x,\forall y,\varphi(x,y) & \exists e,\forall x,\exists t,(\text{TM}_e(x)\downarrow \text{ by } t)\;(\text{uniformity}) \ \hline \Pi_2 & \forall x,\exists y,\varphi(x,y) & \forall x,\exists t,(\text{TM}_e(x)\downarrow \text{ by } t)\;(\text{totality of a fixed } e) \ \hline \Sigma_3 & \exists x,\forall y,\exists z,\varphi(x,y,z) & \exists x,\forall y,\exists z\,(z = x + y)\;(\text{one alternation more}) \ \hline \Pi_3 & \forall x,\exists y,\forall z,\varphi(x,y,z) & \forall x,\exists y,\forall z\,(z \ge y \rightarrow z \ge x)\;(\text{trivial collapse}) \ \hline \end{array}
$$

Upshot. Many discussions implicitly replace a statement of the form

$$
\Pi_2 : \forall x,\exists y, R(x,y)
$$

with something like

$$

$$

$$
\Sigma_3 : \exists f,\forall x,R(x,f(x)).
$$

These are not equivalent in general. Finding *Goldbach's Conjecture* is not the same as finding a *proof* for *Goldbach's Conjecture.* The second statement quantifies over a *uniform policy* $f$. It asserts the existence of a function object, not merely a response for each instance. The arithmetical hierarchy makes this mismatch visible immediately:

Closer look at a few. (i) Start with $\Sigma_n$ formulas whose *prenex* form begins with an existential block

$$
\Sigma_n : \exists \color{green}{x_1}\,\forall \color{blue}{x_2}\,\exists \color{green}{x_3}\color{magenta}{\big(\cdots\big)}
$$

$$
\textsf{Find $\color{green}{\textsf{[method]}}$ for n $\color{blue}{\textsf{[guests]}}$ such that a $\color{green}{\textsf{[guest]}}$ gets $\color{magenta}{\textsf{[1/n pizza]}}$}
$$

This is your “average problem”. There is something that is witnessed by producing an initial object (a certificate, a move, a program, an axiom), after which remaining rounds are obligations against possible challenges. Problem. Nonexistence is hard to establish. We move on.

(ii) Alignment, profitability, robustness against arbitrary inputs. (a) $\Pi_1$ is what all the fuss is about. They are *guarantees* that $x$ provides a response for each challenge, or that at least some appropriate response exists, depending on the challenge. Notoriously hard to prove non-existence:

$$
\forall \color{blue}{x_1}\,\exists \color{green}{x_2}\,\forall \color{blue}{x_3}\color{magenta}{\big(\cdots\big)}
$$

$$
\textsf{Find for all $\color{blue}{\textsf{[ poker turn ]}}$  $\color{green}{\textsf{[ bet ]}}$ such that all $\color{blue}{\textsf{[ oponents ]}}$ $\color{magenta}{\textsf{[ fold ]}}$.}
$$

But helps to frame the actual semantics in syntactic terms:

$$
\textsf{Find for all $\color{blue}{{\textsf{[ tasks ]}}}$ a single $\color{green}{{\textsf{[ AGI ]}}}$ such that all $\color{blue}{{\textsf{[ search is ]}}}$ $\color{magenta}{{\textsf{[ relevant ]}}}$.}
$$

$\Pi_1$ includes foundational theorems, conjectures and problems. *Prime Number theorem*, *Quadratic Reciprocity, Fermat’s Last Theorem.* These mostly “yes-or-no” universal checks: any counterexample would suffice to refute the statement.

Consider,

$$
\color{red}{\textsf{Prove for all [ N > 2 ] there is [ no N ] such that [ $a_1^\textsf{N}+a_2^\textsf{N}+...=a_i^\textsf{N}$ holds ]}}.
$$

$$
\color{red}{\textsf{Absurdum} : }\{\;{27^5 + 84^5 + 110^5 + 133^5 = 144^5\;\}}\vdash\bot.
$$

Universal theorems are brittle and expensive. A single counterexample kills, proofs typically require convincing invariants, induction principles, or uniform constructions.

Next: (b) $\Pi_2$ totalities. They show the limit of understanding, or as always, current understanding.

$$
\forall \color{blue}{x}\,\exists \color{green}{y}\,\varphi\color{magenta}{\big(x,y)}
$$

$$
\textsf{Prove for all } \color{blue}{\textsf{[ even integer N ]}}, \textsf{there are} \color{green}{\textsf{ [ two primes, p and q ]}} \textsf{such that } \color{magenta}{\textsf{[ p + q = N.]}}
$$

These express uniform solvability or totality of a computable functional—obstructive problems or foundational limitations. Almost proven, yet something comes between. e.g. *Riemann hypothesis, Goldbach's conjecture*, *Collatz conjecture*, *Navier Stokes, *$\mathcal{P}\!-\!\mathcal{NP}$.

Interestingly, it's $\Pi_1$-hard to create an AGI, but it's $\Pi_2$-hard to control it (once we understand that alignment itself is witnessed by misalignment).

$$
\textsf{For all $\color{blue}{{\textsf{[ tasks ]}}}$ there are $\color{green}{{\textsf{[ AGI and a specification ]}}}$ such that $\color{magenta}{{\textsf{[ solution ][ is aligned ]}}}$.}
$$

(c)$\Pi_3$ describe the sane uniform limits (aka paradoxes): they quantify over *methods* (thus often over themselves) and their *uniform bounding behavior,  *rather than over individual inputs. *The Diagonal Lemma can be awkwardly expressed as:*

$$
\forall \color{blue}e\color{black},\exists \color{green}n\color{black},\forall \color{blue}m\color{black}\,\Big[\,\color{magenta}{f(n,m)\leftrightarrow f(e,n,m)\color{black }\,}\Big]
$$

$$
\textsf{For all $\color{blue}{\textsf{[ [ [ sentences ]  ] that are a defined as a [ sentence ] ]}}$ }
$$

$$
\textsf{ there exists a }  \color{green}{\textsf{[ sentence that defines ]}}
$$

$$
\textsf{ that any } \color{blue}{\textsf{[ [ sentence ] [ is a [ sentence ] ] ]}}\color{black}{\textsf{ if,}}\color{magenta}{\textsf{ [ and only if, ]}}
$$

$$
\textsf{ $\color{magenta}{\textsf{[  [ there is [  a sentence  ] that defines any [ [ sentence ] ] as such a [ sentence ] ]}}$.}
$$

Note how the diagonal lemma is its own generalization. $\Pi_4$ or higher normal forms trivially collapses to $\Pi_3$.

(iii) At level $\Delta_n$ statements are equivalent (in arithmetic) to both a $\Sigma_n$ and a $\Pi_n$. Encountering them either expected, or the worst case.

$$
\forall u < t \;\; \text{or} \;\; \exists u < t
$$

3.
==

Illustration. We now will equate typical *Chess problems* and *situations* for each.

$$
\Delta_0
$$

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/3bf0f8e2f93be89ec89545afa0a5c6b63f13ce83bbdafa00.png)

“*White moves*.”  
There is one possibility.

We first restrict to possible moves, legal moves, then to beneficial moves under the obligation rule, and finally to the optimal move; here all filters collapse to a single move. Each level is bounded and the answers can be found by a simple search program.

$$
\Sigma_1
$$

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/a8b03f37534e64c97123e87dd282b0dc6e2c2f4ee3fb24b1.png)

“*White* moves: Mate in one.”   
This *deceptive* $\Sigma_1$ problem becomes trivial  
once the problem is *rephrased* in $\Delta_0$.

The above asks for the existence of a single witness move that immediately satisfies a decidable terminal predicate (“is this checkmate?”). Once the actual $\Delta_0$ problem for legality and checkmate are implemented, $\Sigma_1$ reduces to scanning White’s legal moves until one passes the $\Delta_0$ test and a seemingly impossible semantic problem becomes a trivial move.

$$
\Sigma_2
$$

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/0119107041dda7bef8e1ea074a891d3b722bdf08f1ad08b5.png)

“*White* moves: Mate in two”  
White to move and play a move that  
forces checkmate in two, independently of *Black’s* reply*.*

In chess-compositional terms, $\Sigma_2$ could mean *“find a move that places the opponent in Zugzwang or inevitable loss,”* though not yet a full strategy tree.

$$
\Pi_1
$$

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/cd7f481c086b8e67316e84571aa4eeb8fabff09e89118281.png)

“*White* wins.”  
*King-Rook-Rook-King* (KRRK) implies  
that the the winning strategy is preserved

After *White’s* obligation move in KRRK, we quantify over *all* legal Black replies and verify that the same bounded progress predicate still holds (e.g., *Black’s* king remains inside the rook-box, etc.). In other words, $\Pi_1$ means: no matter what Black does next, the constraint survives. A universal filter over replies proves the totality.

$$
\Pi_2
$$

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/1820ced99b3e561e93bf692396819cd3958856b358c871dd.png)

“The *Knight's* Tour”  
*Black* can move such that the *Knight* “visits“  
every square exactly once, regardless where it starts.

The problem above asks: Is it true: for every starting square $s$ (universal choice), there exists at least one move $m$ such that the resulting position admits a completion to a full *Knight's Tour*? Computationally, this is a “for all starts, find a witness move” layer: a universal sweep over squares, with a search that certifies a successful continuation when one exists. Euler settled this question in 1759: Yes.

$$
\Pi_3
$$

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/92657108295afba84602c42b628494cdf68daecca9fec340.png)

“White Moves?”  
The next move is *undecidable*, a fixed point.

This is a *limit case* of rational play, impredicativity stems the self-referential aspect (evaluating moves that preserve equality of evaluation) makes it $\Pi_3$: it quantifies over all positions, all replies, and the meta-property “draw remains draw.”

4.
==

We end with a practical scenario. A management requirement is typically $\Pi_2$-shaped: 

$$
\forall \color{blue}{x}\,\exists \color{green}{y}\,\varphi\color{magenta}{\big(x,y)}
$$

$$
\textsf{For every } \color{blue}{\textsf{[ client request $x$ ]}}, \textsf{there is} \color{green}{\textsf{ [ solution $y$ ]}} \textsf{ such that } \color{magenta}{\textsf{[ client is happy ]}}.
$$

This guarantees solvability instance-by-instance, but it does not commit to a single mechanism. In practice, teams often “upgrade” it into a design claim of the form 

$$
\color{red}{\exists f\,\forall x\,R(x,f(x))}
$$

$$
\color{red}{\textsf{There exists one [ policy ] that [ handles ] all [ requests ]}.}
$$

That “translation" is a stronger commitment, and treating it as equivalent is a reliable way to smuggle new obligations into the spec.

Engineering then tries to push the remaining obligation into a locally auditable core: bounded checks, explicit interfaces, invariants, logs, and certificates where possible—an attempted migration toward $\Delta_n$-style verification conditions even when the surrounding claim remains higher in the hierarchy. Implementation is $\Sigma_n$: produce a witnessand evidence that $R(x,y)$ holds. But after deployment the $\Pi_n$-obligations return: “no input crashes the system,” “no adversarial pattern violates the invariant,” “for every incident there exists a mitigation path.” The pipeline oscillates between $\Pi_n$-claims (global responsibility) and $\Sigma_n$-constructions (deliverables). Confusing $\forall\exists$ with $\exists\forall$is the fastest way to pay twice.

This is why the A*rithmetical Hierarchy* matters. Complexity measures strong properties given a computational model; the hierarchy measures weak logical commitments: what must be exhibited, what must be defended, and how many alternations are built into the claim.

The practical rule is simple: when a statement begins with $\forall$, ask what counts as a counterexample; when it contains $\forall x\,\exists y$, ask whether you are being asked for responses, or for a single uniform policy that produces responses. Read this way, the arithmetical hierarchy becomes a portable diagnostic: a compact language for specifications, proofs, contracts, and systems—anywhere obligations and witnesses trade places.