Category Archives: Gems and stones

Intermediate truth values

I have not written a blog post in a while, so I decided to write up a short observation about truth values in intuitionistic logic which sometimes seems a bit puzzling.

Let $\Omega$ be the set of truth values (in Coq this would be the setoid whose underlying type is $\mathsf{Prop}$ and equality is equivalence $\leftrightarrow$, while in HoTT it is the h-propostions). Call a truth value $p : \Omega$ intermediate if it is neither true nor false, i.e., $p \neq \bot$ and $p \neq \top$. Such a “third” truth value $p$ is proscribed by excluded middle.

The puzzle is to explain how the following two facts fit together:

  1. “There is no intermediate truth value” is an intuitionistic theorem.
  2. There are models of intuitionistic logic with many truth values.

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Constructive gem: juggling exponentials

Constructive gems are usually not about particular results, because all constructive results can be proved classically as well, but rather about the method and the way of thinking. I demonstrate a constructive proof which can be reused in three different settings (set theory, topology, computability) because constructive mathematics has many different interpretations.

Continue reading Constructive gem: juggling exponentials

Constructive gems and stones

In various mathematical forums, mostly those of a logical flavor, I regularly see people asking basic questions about constructive mathematics. I also see misconceptions about constructive mathematics. I shall make a series of posts, Constructive Gems and Stones, which will answer basic questions about constructive mathematics, and will hopefully help fix wrong ideas about constructive mathematics.

A constructive gem is something nice about constructive mathematics that makes you want to know more about it. In contrast, a constructive stone is a complication in constructive mathematics which does not exist in the classical counterpart.

Here we go! The first one is about finite sets.