basic constructions:
strong axioms
further
The weak limited principle of omniscience ($WLPO$) states that the universal quantification of any decidable proposition is again decidable. That is,
Equivalently, that the negation of the existential quantifier of any decidable proposition is decidable:
If one takes the domains of quantification to be subsingletons, one get 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 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 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 $\top$.
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
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 $\top$.
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:
or equivalently
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.
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.
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.
$WLPO$ follows from LPO, but not conversely. If $P(x)$ is a decidable proposition, then so is $\neg{P(x)}$, and so $LPO$ gives
which implies
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.
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)
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)
Created on July 27, 2024 at 15:57:54. See the history of this page for a list of all contributions to it.