# Questions tagged [intuitionism]

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### Reference request for a modification of Bi-Intuitionistic Logic

I’ve asked this same question on math.stackexchange.com, but haven’t received any answers. I ask this question in good faith, so I hope it meets this site’s standards. I have been spending the better ...
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### Limits in free cocompletion, constructively

Classically, if a locally small category $C$ has all limits of shape $K$ (for some small diagram $K$), then its free co-completion also has $K$-shapped limits. But all proof I know of that result ...
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### Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?

Last year, Andrej Bauer gave a talk showing that there is a topos in which the set of Dedekind reals is (sub)countable, and thus, you cannot prove that $\mathbb{R}$ is uncountable without LEM. He ...
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### Preserve validity between the two Kripke frames

The background of our discussion is intuitionistic logic, i.e. the following definitions are intuitionistic Kripke frame. For $n \geq 1$, let $\mathcal{C}_n$ denote the frame which is shown in Fig.1. ...
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### Are there good criteria for the topological models where BD-N and BD hold?

A (non-empty/inhabited) subset $S$ of $\mathbb{N}$ is said to be pseudo-bounded if for every sequence $x_n$ in $S$ we have $\lim_{n\to \infty} \frac{x_n}{n} = 0$ Clearly all bounded subsets are pseudo-...
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### The LNC as a mathematical theorem

One of the most intriguing things I've read about over the last few years is Diaconescu's theorem, which says that, in some forms of constructivist/intuitionistic set theory, even if the law of the ...
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### Is intuitionistic predicate logic (semantically) complete or incomplete?

According to Constructivism in Mathematics: An Introduction by Troelstra A.S. and Van Dalen (https://archive.org/details/constructivismin0002troe/page/718/mode/2up) it is proven in an intuitionisitc ...
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### Markov's principle from constant domain logic

I am looking for a proof of and/or a reference for the result that Markov's principle can be proved in the framework of constant domain logic. By constant domain logic, I mean intuitionistic logic ...
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### Computability-theoretic results relevant to realizability

This may be a very naive question which only reflects my failure at literature search, but: Although realizability (in its original form at least) is grounded in computability, the details of ...
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### Fibers of the morphism from the free Heyting algebra to the free Boolean algebra

For $k\in\mathbb{N}$, let $H_k$ be the free Heyting algebra on $k$ variables $p_1,\ldots,p_k$ and $B_k$ be the free Boolean algebra on the same $k$ variables. Thus, $B_k$ has $2^{2^k}$ elements (...
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### Henkin-style completeness proofs for intuitionistic logic

Henkin-style completeness proofs are founded on a few basic presuppositions, such as the assumptions that the language of a logical theory must be enumerable (or at least that the axiom of choice ...
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### Superintuitionistic logics which are not hereditary/monotonic: impossible or possible?

An intuitionistic Kripke model is a triple $\langle W,\leq, \Vdash \rangle$, where $\langle W,\leq \rangle$ is a preordered Kripke frame, and $\Vdash$ satisfies the following condition of ...
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### Are Regularity schema and $\in$-induction schema equivalent in intuitionistic logic?

In posting "Does Regularity schema imply $\in$-induction when added to first order Zermelo set theor?" the answer was that they are equivalent in classical first order logic with membership "$\in$". ...
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### Going beyond the strength of Peano arithmetic "without sets"

First-order arithmetic is fairly weak, as measured for example by its consistency strength. When a stronger theory is desired, it is common to work with (fragments of) second-order arithmetic or set ...
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### Identity types: What makes Intuitionistic Type Theory *intuitionistic*?

In the opening passage of Martin-Löf's (1975) he famously says that "the theory of types with which we shall be concerned is intended to be a full scale system for formalizing intuitionistic ...
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### Intutionistic Robinson Arithmetic

By Friedman translation $HA$ and $PA$ prove the same $\Pi_2$ formulas. Is it true for Intutionistic Robinson arithmetic(Robinson axioms with intutionistic logic) and classic Robinson arithmetic? ...
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### How strong is Cantor-Bernstein-Schröder?

There are several questions here on MO about the Cantor-Bernstein-Schröder ((C)BS) theorem, but I could not find answers to what arose to me recently. Although I don't think I need to recall it here, ...
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### Did Bishop, Heyting or Brouwer take partial functions seriously?

The partial μ-recursive functions which may or may not be provably total seem to have some direct relation to the initial motivations for intuitionistic mathematics. (Following Kronecker, one ...
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### Forcing is intuitionistic

The main idea of why it´s necessary a generic filter $G$ to extend a countable transitive $\epsilon$-interpretation (not necessarily a model) $M$ is given by the condition (for which $G$ being a ...
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### Models of intuitionistic linear logic that reflect the resource interpretation

I am interested in models of intuitionistic linear logic, that is, the logic that you get if you take classical linear logic and restrict the set of operators to $\otimes$, $1$, $\multimap$, $\times$, ...
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### Is there a probability theory developed in intuitionistic logic?

Since Boole it is known that probability theory is closely related to logic. According to the axioms of Kolmogorov, probability theory is formulated with a (normalized) probability measure \$\mbox{...
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