nLab weak limited principle of omniscience

Definition

The weak limited principle of omniscience ($WLPO$) states that the universal quantification of any decidable proposition is again decidable. That is,

$$(\forall x, P(x) \vee \neg{P(x)}) \Rightarrow (\forall x, P(x)) \vee \neg(\forall x, P(x)).$$

Equivalently, that the negation of the existential quantifier of any decidable proposition is decidable:

$$(\forall x, P(x) \vee \neg{P(x)}) \Rightarrow (\neg \exists x, P(x)) \vee \neg(\neg \exists x, P(x)).$$

We can also state the principle set-theoretically, with explicit reference to the domain of quantification. Given a set $A$, the weak limited principle of omniscience for $A$ ($LPO_A$) states that given a subset $P \in \mathcal{P}(A)$, if $P$ is a decidable subset of $A$ ($P \cup \bar{P} = A$), then either $\bar{P} = A$ or $\neg (P = \emptyset)$, where $\bar{P}$ is the complement of $P$.

If one takes the domains of quantification to be subsingletons, one gets weak excluded middle $\neg p \vee \neg \neg p$ ($WEM$), which is weaker than $EM$; conversely, $WEM$ implies $WLPO$ (over any domain). Again, Bishop's $WLPO$ takes the domain to be $\mathbb{N}$, giving a principle weaker than $WEM$ (and also weaker than $LPO_{\mathbb{N}}$).

In the internal logic

In set theory, there are actually two different notions of logic: there is the external predicate logic used to define the set theory itself, and there is the internal predicate logic induced by the set operations on subsingletons and injections. In particular,

• An internal proposition is a set $P$ such that for all elements $x \in P$ and $y \in P$, $x = y$.

• The internal proposition true is a singleton $\top$.

• The internal proposition false is the empty set

• The internal conjunction of two internal propositions $P$ and $Q$ is the cartesian product $P \times Q$ of $P$ and $Q$.

• The internal disjunction of two internal propositions $P$ and $Q$ is the image of the unique function $!_{P \uplus Q}:P \uplus Q \to 1$ from the disjoint union of $P$ and $Q$ to the singleton $\top$.

$$P \vee Q = \mathrm{im}(!_{P \uplus Q})$$

• The internal implication of two internal propositions $P$ and $Q$ is the function set $P \to Q$ between $P$ and $Q$.

• The internal negation of an internal proposition $P$ is the function set from $P$ to the empty set

$$\neg P = P \to \emptyset$$

• An internal proposition $P$ is a decidable proposition if it comes with a function $\chi_P:P \to 2$ from $P$ to the boolean domain $2$.

• An internal predicate on a set $A$ is a set $P$ with injection $i:P \hookrightarrow A$, whose family of propositions indexed by $x \in A$ is represented by the preimages $i^*(x)$.

• The internal existential quantifier of an internal predicate $i:P \hookrightarrow A$ is the image of the unique function $!_P:P \to \top$ into the singleton $\top$.

$$\exists_A P = \im(!_P)$$

• The internal universal quantifier of an internal predicate $i:P \hookrightarrow A$ is the dependent product of the preimages

$$\forall_A P = \{f \in P^A \vert \forall x \in A, f(x) \in i^*(x) \}$$

• An internal predicate $i:P \hookrightarrow A$ is a decidable proposition if it comes with a function $\chi_P(x):i^*(x) \to 2$ into the boolean domain for all elements $x \in A$, or equivalently if it comes with a function $\chi_P:A \to 2$ from $A$ to the boolean domain $2$.

Then the internal WLPO for a family of sets $(A_z)_{z \in I}$ is the WLPO for each $A_z$ stated in the internal logic of the set theory:

• For any internal predicate $i:P \hookrightarrow A_z$, if there is a function $\chi_P:A_z \to 2$, then the internal universal quantification of $P$, $\forall_{A_z} P = \{f \in P^{A_z} \vert \forall x \in A_z, f(x) \in i^*(x) \}$ has a function $(\forall_{A_z} P) \to 2$ into the boolean domain.

or equivalently

• For any function $a:A_z \to 2$, the internal universal quantification of $P = \{x \in A_z \vert a(x) = 1\}$, $\forall_{A_z} P = \{f \in P^{A_z} \vert \forall x \in A_z, f(x) \in i^*(x) \}$ has a function $(\forall_{A_z} P) \to 2$ into the boolean domain.

The internal versions of the weak limited principle of omniscience, like all internal versions of axioms, are weaker than the external version of the the weak limited principle of omniscience, since while bounded separation implies that one can convert any external predicate $x \in A \vdash P(x)$ on a set $A$ to an internal predicate $\{x \in A \vert P(x)\} \hookrightarrow A$, it is generally not possible to convert an internal predicate to an external predicate without a reflection rule which turns subsingletons in the set theory into propositions in the external logic.

Equivalent statements

There are various other results that are equivalent to the weak limited principle of omniscience. Here are a few:

• The boolean domain is a $\sigma$-frame if and only if $\mathrm{WLPO}_\mathbb{N}$ holds.

• The analytic WLPO for the Cauchy real numbers is equivalent to the WLPO for the natural numbers.

• The untruncated LLPO for the natural numbers is equivalent to the WLPO for the natural numbers.

Truncated and untruncated versions in dependent type theory

In the context of dependent type theory, the weak limited principle of omniscience can be translated in two ways, by interpreting “or” as propositionally truncated (“merely or”) or untruncated (“purely or”). Truncated WLPO and untruncated WLPO are equivalent to each other.

 Related statements

• $WLPO$ follows from LPO, but not conversely. If $P(x)$ is a decidable proposition, then so is $\neg{P(x)}$, and so $LPO$ gives

  $$(\exists x, \neg{P(x)}) \vee (\forall x, \neg{\neg{P(x)}}),$$

  which implies

  $$\neg(\forall x, P(x)) \vee (\forall x, P(x))$$

  as $P$ is decidable.

• The LLPO follows from WLPO, since WLPO is equivalent to untruncated LLPO, which implies truncated LLPO. However, the converse does not necessarily hold, since in http://www1.maths.leeds.ac.uk/~rathjen/Lifschitz.pdf is a model by Michael Rathjen that separates WLPO from LLPO. Similarly, Grossack 24 shows that Johnstone’s topological topos separates WLPO from LLPO.

• The $\mathrm{WLPO}_\mathbb{N}$ implies the stable fan theorem.

Choiceless weak limited principle of omniscience

There is a generalization of the weak limited principle of omniscience defined in King 2024, which was suggested to be called the choiceless weak limited principle of omniscience. The choiceless weak limited principle of omniscience for a set $A$ states that given a set $A$ and a pair of predicates $P$ and $Q$ such that $P(x) \vee Q(x)$ holds for all $x \in A$, then either for all $x \in A$, $Q(x)$, or it is not true that for all $x \in A$, $\neg P(x)$.

$$(\forall x \in A.P(x) \vee Q(x)) \Rightarrow ((\forall x \in A.Q(x)) \vee \neg (\forall x \in A.\neg P(x)))$$

The idea behind the term “choiceless” is that one isn’t forced to choose between $P(x)$ or $Q(x)$ in this version of the weak limited principle of omniscience. One gets back the usual weak limited principle of omniscience if $P(x)$ and $Q(x)$ are mutually exclusive propositions for all $x \in A$, from which $Q(x)$ if and only if $\neg P(x)$ for all $x \in A$.

We can also state the principle set-theoretically. Given a set $A$, the choiceless weak limited principle of omniscience for $A$ states that given subsets $P, Q \in \mathcal{P}(A)$, if $P \cup Q = A$, then either $Q = A$ or $\neg (P = \emptyset)$. One gets back the usual limited principle of omniscience if $P$ and $Q$ are disjoint subsets $P \cap Q = \emptyset$, from which $Q = \bar{P}$, where $\bar{P}$ is the complement of $P$.

The choiceless weak limited principle of omniscience for the natural numbers $\mathbb{N}$ implies the analytic weak limited principle of omniscience for all sets of real numbers, as shown in King 2024.

Related concepts

• principle of omniscience

• limited principle of omniscience

• lesser limited principle of omniscience

• Markov's principle

• weak excluded middle

 References

• Errett Bishop: Foundations of Constructive Analysis, McGraw-Hill (1967)

  (in the introduction or chapter 1, I forget)

• Hajime Ishihara, Reverse Mathematics in Bishop’s Constructive Mathematics, Philosophia Scientiæ, CS 6 (2006) (doi:10.4000/philosophiascientiae.406, pdf)

• Mike Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)

• Hannes Diener, Constructive reverse mathematics, Habilitationsschrift Fakultät IV - Naturwissenschaftlich-Technische Fakultät, 2018. (arXiv:1804.05495, dspace:ubsi/1306)

• Auke Booij, Analysis in univalent type theory (2020) [etheses:10411, pdf]

• Madeleine Birchfield, Andrew Swan (2024) on Category Theory Zulip, LPO and sigma-frame structure on booleans

• Chris Grossack, Life in Johnstone’s Topological Topos 3 – Bonus Axioms (web)

• Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, A Foundation for Synthetic Stone Duality (arXiv:2412.03203)

• Christopher King, What are these generalizations of the principles of omniscience called?, MathOverflow, 15 February 2024. (web)