# Mathematics and Computation

## A blog about mathematics for computers

Postsby categoryby yearall

# Posts in the category 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.

### Constructive gem: an injection from Baire space to natural numbers

I am not sure whether to call this one a constructive gem or stone. I suppose it is a matter of personal taste. I think it is a gem, albeit a very unusual one: there is a topos in which $\mathbb{N}^\mathbb{N}$ can be embedded into $\mathbb{N}$.

### Constructive gem: double exponentials

In the last constructive gem we studied the exponential $2^\mathbb{N}$ and its isomorphic copies. This time we shall compute the double exponential $2^{2^\mathbb{N}}$ and even write some Haskell code.

### 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.

### Constructive stone: minima of sets of natural numbers

I promise I will post a constructive gem soon. This constructive stone came up as a reaction to the cardinality of finite sets stone. I show that inhabited sets of natural numbers need not have minima, constructively.

### Constructive stone: cardinality of sets

Cardinality of sets in constructive mathematics is not as well behaved as in classical mathematics. Cardinalities of finite sets are not natural numbers, and cardinalities are not linearly ordered.

### Constructive stone: finite sets

Just like in real life, constructive stones are easier to find than constructive gems, so let me start the series with a stone about constructive finite sets.

Two girl one cup