nLab principle of omniscience

Idea

In logic and foundations, a principle of omniscience is any principle of classical mathematics that is not valid in constructive mathematics. The idea behind the name (which is due to Bishop (1967)) is that, if we attempt to extend the computational interpretation of constructive mathematics to incorporate one of these principles, we would have to know something that we cannot compute. The main example is the law of excluded middle ($EM$); to apply $p \vee \neg{p}$ computationally, we must know which of these disjuncts hold; to apply this in all situations, we would have to know everything (hence ‘omniscience’).

Bishop's principles of omniscience (stated below) may be seen as principles that extend classical logic from predicates (where $EM$ may happen to be valid, even constructively, for certain predicates) to their quantifications over infinite domains (where $EM$ is typically not constructively valid).

Definition

The limited principle of omniscience

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

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

or equivalently

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

We have not stated the domain of quantification of the variable $x$. If you take it to be the subsingleton corresponding to a given truth value and apply this principle to the constantly true proposition, then $EM$ follows; conversely, $EM$ implies $LPO$ (over any domain). However, Bishop's $LPO$ takes the domain to be the set of natural numbers, giving a weaker principle than $EM$. (It appears that a realizability topos based on infinite-time Turing machines validates $LPO$ but not $EM$; see Bauer (2011).) Note that propositions of the form $\exists x, P(x)$ when $P$ is decidable are the open truth values in the Rosolini dominance.

We can also state the principle set-theoretically, with explicit reference to the domain of quantification. Given a set $A$, the limited principle of omniscience for $A$ ($LPO_A$) states that, given any function $f$ from $A$ to the boolean domain $\{0,1\}$, either $f$ is the constant map to $0$ or $1$ belongs to the image of $f$. Then Bishop's $LPO$ is $LPO_{\mathbb{N}}$, which applies to any infinite sequence of bits.

While $LPO$ for $\mathbb{N}$ is a classic example of the difference between constructive and classical mathematics, $LPO$ holds for the set $\overline{\mathbb{N}}$ of extended natural numbers; this is related to the fact that $\overline{\mathbb{N}}$ may constructively be given a compact topology. See Escardó (2011) for this and much more.

The weak limited principle of omniscience

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

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

The lesser limited principle of omniscience

The lesser limited principle of omniscience ($LLPO$) 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, as before, you take the domains of quantification to be subsingletons, you 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 $LPO_{\mathbb{N}}$).

Stated set-theoretically, 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 $LLPO_{\mathbb{N}}$.

Relation between the principles of omniscience

We have the following relations between the three principles of omniscience:

• $WLPO$ 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.

• $LLPO$ follows from $LPO$, but not conversely.

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 $2$.

Then the internal LPO for a family of sets $(A_z)_{z \in I}$ is the LPO 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 existential quantification of $P$, $\exists_{A_z} P = \im(!_P)$ has a function $(\exists_{A_z} P) \to 2$ into the boolean domain.

or equivalently, as

• For any function $a:A_z \to 2$, the internal existential quantification of $P = \{x \in A_z \vert a(x) = 1\}$, $\exists_{A_z} P = \im(!_P)$ has a function $(\exists_{A_z} P) \to 2$ into the boolean domain.

Similarly, 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.

And finally, 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 LPO for the family of all subsingletons is internal excluded middle and the internal LLPO for the family of all subsingletons is internal weak excluded middle. Similarly, given a universe $U$, the internal LPO for the family of all sets in $U$ is excluded middle in $U$, and the internal LLPO for the family of all sets in $U$ is weak excluded middle in $U$.

The internal versions of the principles of omniscience, like all internal versions of axioms, are weaker than the external version of the 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.

Truncated and untruncated versions in dependent type theory

In the context of dependent type theory, the various 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 principles of omniscience are as follows are:

• Truncated LPO and untruncated LPO are equivalent (due to Martin Escardo, see UFP)

• Similarly, truncated WLPO and untruncated WLPO are equivalent.

• 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 LLPO. Similarly, Grossack 24 shows that Johnstone’s topological topos separates WLPO from LLPO.

Analytic principles of omniscience

Bishop introduced the above principles of omniscience to show that certain results in pointwise analysis could not be constructive, by showing that these results implied a principle of omniscience. There are similar axioms in analysis which imply the principles of omniscience for natural numbers, called analytic principles of omniscience, and include the following:

• The analytic LPO states that the usual apartness relation on the set $\mathbb{R}$ of real numbers is decidable ($x \neq y$ or $x = y$), or equivalently trichotomy for the real numbers ($x \lt y$ or $x = y$ or $x \gt y$), or equivalently, that the real numbers form a discrete field.

• The analytic WLPO states that $\mathbb{R}$ has decidable equality, or that the partial order on the real numbers is decidable ($x \leq y$ or $\neg (x \leq y)$).

• The analytic LLPO states that the usual order on $\mathbb{R}$ is a total order ($x \leq y$ or $x \geq y$), which (by analogy with trichotomy) may be called dichotomy for the real numbers.

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).

• Every semi-decidable proposition is a decidable proposition iff $\mathrm{LPO}_\mathbb{N}$ holds.

• Sequential compactness of the unit interval holds if and only if $\mathrm{LPO}_\mathbb{N}$ holds.

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

• The Cauchy real numbers are isomorphic to the radix expansion in any base (e.g., a decimal expansion or binary expansion) iff $\mathrm{LPO}_\mathbb{N}$ holds. See Feldman (2010).

• Various analytic principles of omniscience are equivalent to the principles of omniscience for the natural numbers (see analytic principles of omniscience for proofs). More specifically:

  • The analytic LPO for the following sets of real numbers are equivalent to the LPO for the natural numbers: the Cauchy real numbers $\mathbb{R}_C$, the Escardo-Simpson real numbers/HoTT book real numbers $\mathbb{R}_E$/$\mathbb{R}_H$, and the subfield of Dedekind real numbers $\mathbb{R}_\Sigma \subseteq \mathbb{R}_D$ which are constructed out of Dedekind cuts valued in the initial $\sigma$-frame $\Sigma \subseteq \Omega$.

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

  • 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:

• 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).

• The full bar theorem implies the $\mathrm{LPO}_\mathbb{N}$.

• The existence of various classical universes or models of foundations of mathematics implies the $\mathrm{LPO}_\mathbb{N}$:

  • Any model $\mathcal{V}$ of bounded Zermelo set theory contains a pure set of real numbers $\mathbb{R}$. One can collect all the pure sets of $\mathcal{V}$ which are in $\mathbb{R}$ and show that the resulting set is a subset of $\mathcal{V}$ and a sequentially Cauchy complete Archimedean ordered field which satisfies the analytic LPO, thus implying $\mathrm{LPO}_\mathbb{N}$ for the entire foundations. Thus, the existence of stronger models of material set theory such as ZFC also imply $\mathrm{LPO}_\mathbb{N}$ for the entire foundations.

  • For a similar reason, the existence of a constructively well-pointed Boolean W-topos $\mathcal{E}$ implies the $\mathrm{LPO}_\mathbb{N}$, since the hom-set $\mathrm{Hom}_\mathcal{E}(1, \mathbb{R})$, where $1 \in \mathcal{E}$ is the terminal generator and $\mathbb{R} \in \mathcal{E}$ is the real numbers object in $\mathcal{E}$, yields a sequentially Cauchy complete Archimedean ordered field which satisfies the analytic LPO, thus implying $\mathrm{LPO}_\mathbb{N}$ for the entire foundations. Thus, the existence of any constructive model of ETCS also implies $\mathrm{LPO}_\mathbb{N}$ for the entire foundations.

  • Finally, in dependent type theory, there being a univalent Tarski universe $(U, T)$ closed under dependent product types, dependent sum types, and identity types and satisfying the axiom of infinity and excluded middle implies the $\mathrm{LPO}_\mathbb{N}$, since one can construct an element $\mathbb{R}:U$ representing the $U$-small type of real numbers, whose type reflection $T(\mathbb{R})$ is a sequentially Cauchy complete Archimedean ordered field which satisfies the analytic LPO, thus implying $\mathrm{LPO}_\mathbb{N}$ for the entire type theory. Thus, any univalent Tarski universe which has axiom of choice or a choice operator for the types in the universe also implies $\mathrm{LPO}_\mathbb{N}$ for the entire type theory.

  • Note that in all these cases, the real numbers $\mathbb{R}$ constructed from these universes or classical models of foundations of mathematics, while equivalent to the internal Dedekind real numbers constructed in the universe or model, are not necessarily equivalent to the external Dedekind real numbers in the foundations.

Models

• Assuming that Set is a Boolean topos, then $LPO_{\mathbb{N}}$ (the LPO for natural numbers) holds in any presheaf topos over $Set$ and indeed in any locally connected topos over $Set$, essentially since then $2^N$ is a constant object.

• The LPO for natural numbers fails in Johnstone’s topological topos, due to its internal continuity principle. Hence, the analytic LPO also fails, since the modulated Cauchy reals and Dedekind reals coincide in this topos. However, the (analytic) LLPO holds, as a consequence of the preservation of finite closed unions by the inclusion of sequential spaces.

Related concepts

• Markov's principle

• analytic principle of omniscience

References

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

  (in the introduction or chapter 1, I forget)

• Fred Richman, Polynomials and linear transformations. Linear Algebra and its Applications, Volume 131, 1 April 1990, Pages 131-137. [doi:10.1016/0024-3795(90)90379-Q]

• 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

• Andrej Bauer (2011) via constructivenews, An Injection from $\mathbb{N}^{\mathbb{N}}$ to $\mathbb{N}$ (pdf)

• Martín Escardó (2011) via constructivenews, Infinite sets that satisfy the principle of omniscience in all varieties of constructive mathematics (pdf)

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

• 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)

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

• 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]

• Francesco Ciraulo, $\sigma$-locales in Formal Topology, Logical Methods in Computer Science, Volume 18, Issue 1 (January 12, 2022) (doi:10.46298/lmcs-18%281%3A7%292022, arXiv:1801.09644)

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

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

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

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

• Henri Lombardi, Assia Mahboubi, Théories géométriques pour l’algèbre des nombres réels sans test de signe ni axiome de choix dépendant (arXiv:2406.15218)