Articles of constructive mathematics

Inequality and Intuitionistic logic

$x, y \in \mathbb{R}$ Is the proposition $x \leq y \Rightarrow x=y \lor x<y$ true in intuitionistic logic ? And what about $x \leq y \Rightarrow \lnot(\lnot(x=y \lor x<y))$ (with $\lnot$ the negation) Thanks for your help !

Proving “Or Statements” in Type Theory

Oftentimes in math we see statements of the form $P \to (Q \vee R)$. To prove them we can assume $P$ is true and $R$ is false, and then demonstrate that $Q$ is true. This method of proof has the form: $$ [ (P \wedge \neg R) \to Q ] \to [ P \to ( […]

Intuitionistic proof of $\forall x(P(x)\lor Q(x))\to(\forall x P(x)\lor\exists x Q(x))$

Similar to Is $ \forall x(P(x) \lor Q(x)) \vdash \forall x P(x) \lor \exists xQ(x) $ provable?, but with intuitionistic logic. I expect it is not, since I don’t think the $\exists x Q(x)$ on the right is very “constructive”, but curiously the finite version of this $$((p_1\lor q_1)\land(p_2\lor q_2))\to((p_1\land p_2)\lor(q_1\lor q_2))$$ is intutitionistically provable, […]

Axiom of Choice, Continuity and Intermediate Value Theorem

I am trying to understand a proof I read in Herrlich’s book Axiom of Choice. For those who know the book, it is theorem 4.54 on page 74. The part I am interested in reads: (9) A function $f:X \rightarrow \mathbb{R}$, defined on some subspace $X$ of $\mathbb{R}$, is continuous iff it is sequentially continuous. […]

Formal notion of computational content

In constructive mathematics we often hear expressions such as “extracting computational content from proofs”, “the constructivity of mathematics lies in its computational content”, “realizability models allow to study the computational content”, “intuitionistic double-negation forgets the computational content of a proposition”, and so on. I was wondering if there is a formal notion of computational content […]

Is there a simple example of how the law of the excluded middle can be inapplicable?

Why does a logic system not use the law of the excluded middle? I studied non-classical logic (intuitionistic and modal) where double negation can’t be removed and the law of excluded middle can’t be used. But what is a simple example that non-mathematicians can directly understand where we can’t use the law of the excluded […]

Is ¬¬(¬¬P → P) provable in intuitionistic logic?

I have a feeling it’s not, because ¬¬P → P is not provable. If it is, I’m not sure what kind of reductio I’d need to negate ¬(¬¬P → P). I believe a textbook somewhere said it was provable in intuitionistic logic, so am I missing something or is the textbook wrong?

Does the law of the excluded middle imply the existence of “intangibles”?

First off, I’m not sure if “intangible” is standard terminology, Wikipedia defines an intangible object to be: “objects that are proved to exist, but which cannot be explicitly constructed”. So if someone could point me towards better terminology, I’d appreciate it. The linked article from Wikipedia claims that the axiom of choice implies the existence […]

constructive canonical form of orthogonal matrix

For every orthogonal matrix $Q$ over the reals there is an orthogonal matrix $P$ and a block diagonal matrix $D$ such that $D=PQP^{t}$. Each block in D is either $(1)$, $(-1)$ or a two dimensional block of the form $\left( \begin{array}{cc} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \\ \end{array} \right) $. Is there a […]

How can some statements be consistent with intuitionistic logic but not classical logic, when intuitionistic logic proves not not LEM?

I’ve heard that some axioms, such as “all functions are continuous” or “all functions are computable”, are compatible with intuitionistic type theories but not their classical equivalents. But if they aren’t compatible with LEM, shouldn’t that mean they prove not LEM? But not LEM means not (A or not A) which in particular implies not […]