nLab lesser limited principle of omniscience

Definition

The lesser limited principle of omniscience ($LLPO$) is a principle of omniscience which states that, if the existential quantification of the conjunction of two decidable propositions is false, then one of their separate existential quantifications is false. That is,

$$(\forall x, P(x) \vee \neg{P(x)}) \Rightarrow (\forall y, Q(y) \vee \neg{Q(y)}) \Rightarrow \neg(\exists x, P(x) \wedge \exists y, Q(y)) \Rightarrow \neg(\exists x, P(x)) \vee \neg(\exists y, Q(y)) ,$$

or equivalently

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

If one takes the domains of quantification to be subsingletons, one get de Morgan's law $\neg(p \wedge q) \Rightarrow \neg{p} \vee \neg{q}$ ($DM$), which is weaker than $EM$; conversely, $DM$ implies $LLPO$ (over any domain). Again, Bishop's $LLPO$ takes the domain to be $\mathbb{N}$, giving a principle weaker than $DM$ (and also weaker than $LPO_{\mathbb{N}}$).

We can also state the principle set-theoretically, with explicit reference to the domain of quantification. Given a set $A$, the lesser limited principle of omniscience for $A$ ($LLPO_A$) states that given subsets $P, Q \in \mathcal{P}(A)$, if $P$ and $Q$ are decidable subsets of $A$ ($P \cup \bar{P} = A$, $Q \cup \bar{Q} = A$), then if it is not true that both $P$ and $Q$ are inhabited subsets of $A$, then either $\bar{P} = A$ or $\bar{Q} = A$, where $\bar{P}$ and $\bar{Q}$ are the complements of $P$ and $Q$ respectively.

One can also use functions to the boolean domain instead of decidable subsets: the lesser limited principle of omniscience for $A$ ($LLPO_A$) states that, given functions $f, g\colon A \to \{0,1\}$, if $1 \notin \im f \cap \im g$, then $1 \notin \im f$ or $1 \notin \im g$. So Bishop's $LLPO$ is $LLPO_{\mathbb{N}}$.

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 $x \in A$ and $y \in A$, $x = y$.

• 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 $2$.

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

• The internal LPO for a family of sets $(A_z)_{z \in I}$ holds only for the internal predicates $i:P \hookrightarrow A_z$ which comes with an internal predicate $j:Q \hookrightarrow A_z$ such that $i^*(x) = \neg j^*(x)$ for all $x \in A_z$.

In particular, the internal LLPO for the family of all sets in $U$ is weak excluded middle in $U$.

The internal LLPO, like all internal versions of axioms, are weaker than the external version of LLPO, 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 principles of omniscience. Here are a few:

• Let $[0,1]/(0 \sim 1)$ be the quotient of the unit interval that identifies the endpoints, and let $\mathbb{R}/\mathbb{Z}$ be the quotient ring; both are classically isomorphic to the circle $\mathbb{S}^1$. (Constructively, we take $\mathbb{S}^1$ to be $\mathbb{R}/\mathbb{Z}$, although $S^1$ can also be constructed as a completion of $[0,1]/(0 \sim 1)$.) Constructively, there is an injection $[0,1]/(0 \sim 1) \hookrightarrow \mathbb{R}/\mathbb{Z}$, which is a bijection if and only if the $LLPO$ holds (for the appropriate kind of real number).

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

 Related statements

There are various other results that are related to the principles of omniscience. Here are a few:

• The limited principle of omniscience $\mathrm{LPO}_\mathbb{N}$ and the weak limited principle of omniscience $\mathrm{WLPO}_\mathbb{N}$ for natural numbers imply $\mathrm{LLPO}_\mathbb{N}$, but not conversely.

• The lesser limited principles of omniscience implies that every pointwise continuous function from the Cauchy unit interval to the Cauchy real numbers is a uniformly continuous function. See section 6.1 of Diener (2018).

• In the presence of weak countable choice, there exists a radix expansion in any base (e.g., a decimal expansion or binary expansion) for every Cauchy real number iff $\mathrm{LLPO}_\mathbb{N}$ holds. Without weak countable choice, Lifschitz realizability gives a model in which $\mathrm{LLPO}_\mathbb{N}$ holds but it is not true that there exists a radix expansion in any base for every Cauchy real number. See Swan (2024).

• In the presence of countable choice, $\mathrm{LLPO}_\mathbb{N}$ is equivalent to the claim that the rings of radix expansions in any two bases are isomorphic. See Daniel Mehkeri's answer to Feldman (2010).

Untruncated LLPO in dependent type theory

In the context of dependent type theory, the lesser limited principles of omniscience can be translated in two ways, by interpreting “or” as propositionally truncated (“merely or”) or untruncated (“purely or”). The relationships between the truncated and untruncated LLPO are as follows are:

• Untruncated LLPO is equivalent to WLPO (also due to Martin Escardo).

• In http://www1.maths.leeds.ac.uk/~rathjen/Lifschitz.pdf is a model by Michael Rathjen that separates WLPO from (truncated) LLPO. Similarly, Grossack 24 shows that Johnstone’s topological topos separates WLPO from (truncated) LLPO.

Thus, untruncated LLPO is strictly stronger than truncated LLPO, unlike the case for LPO and WLPO.

Models

• The (analytic) LLPO holds in Johnstone’s topological topos, as a consequence of the preservation of finite closed unions by the inclusion of sequential spaces.

Choiceless lesser limited principle of omniscience

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

$$((\forall x \in A.P(x) \vee Q(x)) \wedge (\forall x \in A.R(x) \vee S(x))) \Rightarrow \neg (\exists x \in A.P(x) \wedge \exists x \in A.R(x)) \Rightarrow ((\forall x \in A.Q(x)) \vee (\forall x \in A.S(x)))$$

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

We can also state the principle set-theoretically, with explicit reference to the domain of quantification. Given a set $A$, the choiceless lesser limited principle of omniscience for $A$ states that given subsets $P, Q, R, S \in \mathcal{P}(A)$, if $P \cup Q = A$ and $R \cup S = A$), then if it is not true that both $P$ and $R$ are inhabited subsets of $A$, then either $Q = A$ or $S = A$. One gets back the usual lesser limited principle of omniscience if $P$ and $Q$ are disjoint subsets $P \cap Q = \emptyset$ and $R$ and $S$ are disjoint subsets $R \cap S = \emptyset$, from which $Q = \bar{P}$ and $S = \bar{R}$, where $\bar{P}$ and $\bar{S}$ are the complements of $P$ and $S$ respectively.

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

Related concepts

• principle of omniscience

• limited principle of omniscience

• weak limited principle of omniscience

• Markov's principle

• de Morgan's law

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)

• David Feldman (2010) on Math.Overflow, Radix notation and toposes

• Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)

• Michael Rathjen, Constructive Zermelo-Fraenkel set theory and the limited principle of omniscience. [arXiv:1302.3037]

• 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)

• Michael Rathjen, Andrew Swan, Lifschitz Realizability as a Topological Construction. The Journal of Symbolic Logic, Volume 85, Issue 4, December 2020, pp. 1342 - 1375. [doi:10.1017/jsl.2021.1, arXiv:1806.10047]

• Andrej Bauer, James Hanson, The Countable Reals (arXiv:2404.01256)

• Andrew Swan (2024) on Category Theory Zulip, Radix expansions in constructive mathematics

• 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)