Bounded inductive dichotomy: separation of open and clopen determinacies with finite alternatives in constructive contexts

Abstract

In his previous work, the author has introduced the axiom schema of inductive dichotomy, a weak variant of the axiom schema of inductive definition, and used this schema for elementary ($\varDelta ^1_0$) positive operators to separate open and clopen determinacies for those games in which two players make choices from infinitely many alternatives in various circumstances. Among the studies on variants of inductive definitions for bounded ($\varDelta ^0_0$) positive operators, the present article investigates inductive dichotomy for these operators, and applies it to constructive investigations of variants of determinacy statements for those games in which the players make choices from only finitely many alternatives. As a result, three formulations of open determinacy, that are all classically equivalent with each other, are equivalent to three different semi-classical principles, namely Markov’s Principle, Lesser Limited Principle of Omniscience and Limited Principle of Omniscience, over a suitable constructive base theory that proves clopen determinacy. Open and clopen determinacies for these games are thus separated. Some basic results on variants of inductive definitions for $\varDelta ^0_0$ positive operators will also be given.

A relation to Veldman’s determinacy

As mentioned in Introduction, Veldman [24] introduced the notion of I-determinateness, which is senseful in the intuitionistic context, by replacing the negated premise with its “strong negation” from our determinacy implication. Namely, his formulation is as follows.

($\mathcal {C}\mathsf{-VelDet} ^*$):

$(\forall \tau \,{<}\,\underline{2})(\exists \pi \,{<}\,\underline{2}) (\mathrm {Ob2}(\pi ,\tau )\,{\wedge }\,\varphi (\pi ))\, {\rightarrow }\,(\exists \sigma \,{<}\,\underline{2})\mathrm {WS1}^*_\varphi (\sigma )$,

for any $\mathcal {C}$ formula $\varphi (\pi )$ with a distinguished variable $\pi$.

He also defined the notion of II-determinateness by exchanging the players but not replacing $\varphi$ by $\lnot \varphi$ as Player II’s winning condition (as opposed to our $\mathcal {C}\mathsf{-DetImp2} ^*$), which turns to be equivalent to $\mathcal {C}\mathsf{-VelDet} ^*$ for reasonable $\mathcal {C}$, by a similar argument as Lemma 5.

Although Veldman admitted that strong negations must be defined manually on a case-by-case basis and cannot be defined systematically, for our purpose $(\forall \pi \,{<}\,\underline{2})\exists x \lnot \varphi (\pi {\upharpoonright }x)$ seems plausible to be called the strong negation of $\mathrm {WK2}_{\varphi }$. Thus, we can expect that $\varSigma ^0_1\mathsf{-VeDet1} ^*$ is equivalent to $\varDelta ^0_0\mathsf{-WFT}$. Actually, this equivalence was proved by Veldman [24] over his base theory, which has full induction scheme and hence much stronger than ours $\mathbf {EL}_0^-$. By modifying our previous proofs, we can show that the equivalence is provable in $\mathbf {EL}_0^-$.

Lemma 17

$\varSigma ^0_1\mathsf{-VelDet} ^*$ and $\varDelta ^0_0\mathsf{-WFT}$ are equivalent over $\mathbf {EL}_0^-$.

Proof

Our strategy is to modify the proofs of Lemmata 7, 8 (2) and 11 so that implications among the “strong negations” of the second disjuncts (in the reverse direction), where as the “strong negation” of $\mathrm {SIDic2}_\varphi (x)$ we take

$$\begin{aligned} \forall F\exists y (x\,{\in }\,F\,{\vee }\, (\varphi (y,F)\,{\wedge }\,y\,{\notin }\,F)). \end{aligned}$$

First, to modify Lemma 7, assuming this, for any $\pi \,{<}\,\underline{2}$ we show $\exists y \lnot \psi (x,\pi {\upharpoonright }y)$, where $\psi$ is as in Lemma 7. Let $F\,{:=}\, \{y\,{:}\,\pi (y)\,{=}\,0\}$. By assumption, either $\pi (x)\,{=}\,0$ and so $\lnot \psi (x,\pi {\upharpoonright }(x{+}1))$ or $\lnot \psi (x,\pi {\upharpoonright }(t(y){+}1))$ for some y, where t is as given in the proof of Lemma 7.

Next, to modify Lemma 8 (2), assume $(\forall \tau \,{<}\,\underline{2})(\exists \pi \,{<}\,\underline{2}) (\mathrm {Ob2}(\pi ,\tau )\,{\wedge }\,\varphi (\pi ))$. For any F, define $\tau \,{<}\,\underline{2}$ as in Lemma 8 (2). The assumption yields u with $\mathrm {Ob2fr}(u,\tau ,\langle \,\rangle ) \,{\wedge }\,\theta (u)$. If $u{\upharpoonright }0\,{=}\, \langle \,\rangle \,{\notin }\,F$ and $(\forall y\,{\le }\,|u|) (\mathrm {DetO}[\theta ](u{\upharpoonright }y,F)\,{\rightarrow }\, u{\upharpoonright }y\,{\in }\,F)$, then $u{\upharpoonright }y\,{\notin }\,F\,{\wedge }\, \lnot \theta (u{\upharpoonright }y)$ by induction on y, contradicting $\theta (u)$. Thus, either $\langle \,\rangle \,{\in }\,F$ or $(\exists y\,{\le }\,|u|) (\mathrm {DetO}[\theta ](u{\upharpoonright }y,F)\,{\wedge }\, u{\upharpoonright }y\,{\notin }\,F)$.

Finally to modify Lemma 11, assume $(\forall \pi \,{<}\,\underline{2}) \exists x\lnot \varphi (\pi {\upharpoonright }x)$, where we may assume $\varphi (\langle \,\rangle )$. For any $\tau \,{<}\,\underline{2}$, define $\rho \,$ as in the proof of Lemma 11, and $\pi \,{<}\,\underline{2}$ so that $\pi (y)\,{=}\,\tau (\rho (\pi \upharpoonright y))$ for any y. The assumption yields x with $\lnot \varphi (\pi {\upharpoonright }(x{+}1))$, and we may assume $(\forall y\,{\le }\,x)\varphi (\pi {\upharpoonright }y)$. By induction on $z\,{\le }\,x{+}1$, we now show the following, which is equivalent to a $\varDelta ^0_0$ formula:

$$\begin{aligned}&(\exists y\,{<}\,z)\exists u \begin{pmatrix}\mathrm {Bin}(u,y) \,{\wedge }\,\tilde{\varphi }((\pi {\upharpoonright }(x{-}y)){*}\langle 1{-}\pi (x{-}y)\rangle {*}u) \\ {\wedge }\,\forall v(\mathrm {Bin}(v,y)\,{\rightarrow }\,\lnot \tilde{\varphi }(\pi {\upharpoonright }(x{+}1{-}y)){*}v) \end{pmatrix} \\&\,\,{\vee }\; \forall v \begin{pmatrix} \mathrm {Bin}(v,z)\,{\rightarrow }\lnot \tilde{\varphi }((\pi {\upharpoonright }(x{+}1{-}z)){*}v) \end{pmatrix}. \end{aligned}$$

If $z\,{=}\,0$, the latter disjunct is obviously the case. For $z\,{>}\,0$, if the former is the case for $z{-}1$ then it is the case for z. So assume $\forall v(\mathrm {Bin}(v,z{-}1)\,{\rightarrow }\,\lnot \tilde{\varphi }((\pi {\upharpoonright }(x{+}2{-}z)){*}v))$ with $z\,{>}\,0$. If $\forall v(\mathrm {Bin}(v,z)\,{\rightarrow }\,\lnot \tilde{\varphi }((\pi {\upharpoonright }(x{+}1{-}z)){*}v))$ is not the case, then by $\varDelta ^0_0\mathsf{-LEM}$ we have u with $\mathrm {Bin}(u,z{-}1)\,{\wedge }\, \tilde{\varphi }((\pi {\upharpoonright }(x{+}1{-}z)) {*}\langle 1{-}\pi (x{-}z)\rangle {*}u))$, and hence the pair of this u and $y\,{:=}\,z{-}1$ witnesses the former disjunct for z. Once the induction is completed, with $z\,{=}\,x{+}1$, we have

$$\begin{aligned}&(\exists y\,{\le }\,x)\exists u\begin{pmatrix}\mathrm {Bin}(u,y) \,{\wedge }\,\tilde{\varphi }((\pi {\upharpoonright }(x{-}y)) {*}\langle 1{-}\pi (x{-}y)\rangle {*}u) \,{\wedge }\\ \forall v(\mathrm {Bin}(v,y)\,{\rightarrow }\, \lnot \tilde{\varphi }((\pi {\upharpoonright }(x{-}y)) {*}\langle \pi (x{-}y)\rangle {*}v)) \end{pmatrix}\\&\quad \quad \;\;\;\;{\vee }\; \forall v(\mathrm {Bin}(v,x{+}1)\,{\rightarrow }\,\lnot \tilde{\varphi }(v)). \end{aligned}$$

In the former case, $\mathrm {Ob2}(\pi ',\tau )\,{\wedge }\,\mathrm {WKG}[\varphi ](\pi ')$ for a play $\pi '$ in which Player I chooses $\pi {\upharpoonright }(x{-}y)$ in the first phase and u in the third phase. In the latter case, $\mathrm {Ob2}(\pi '',\tau )\,{\wedge }\,\mathrm {WKG}[\varphi ](\pi '')$ for a play $\pi ''$ in which Player I chooses $\pi {\upharpoonright }x$ in the first phase. $\square$

The last part of the proof clarifies the connection between the main disjunction of our game $\mathrm {WKG}[\varphi ]$ and the disjunctive formula to which induction is applied in [24, Lemma 5.3].

On the other hand, we do not know the strength of the result of applying the same modification to $\varSigma ^0_1\mathsf{-DetImp2} ^*$: for any $\varSigma ^0_1$ formula $\varphi (\pi )$ with a distinguished variable $\pi$,

$$\begin{aligned} (\forall \sigma \,{<}\,\underline{2})(\exists \pi \,{<}\,\underline{2}) (\mathrm {Ob1}(\sigma ,\pi )\,{\wedge }\,\lnot \varphi (\pi ))\, {\rightarrow }\,(\exists \tau \,{<}\,\underline{2})\mathrm {WS2}^*_\varphi (\tau ). \end{aligned}$$

Similarly to the comment concerning the first question raised at the end of the last section, we can point out that some mystery happens for the first disjuncts, but that the situation for the second disjuncts is rather straightforward.