Homotopy Type Theory UMyn8W7b (Rev #34)

On foundations

The natural numbers are characterized by their induction principle (in second-order logic/in a higher universe/as an inductive type). If one only has a first order theory, then one cannot have an induction principle, and instead one has a entire category of models. Thus, the first order models of arithmetic typically found in classical logic and model theory do not define the natural numbers, and this is true even of first-order Peano arithmetic.

Real numbers

Lattices and $\sigma$-frames

A lattice is a set $L$ with terms $\bot:L$, $\top:L$ and functions $\wedge:L$, $\vee:L$, such that

• $(L, \top, \wedge)$ and $(L, \bot, \vee)$ are commutative monoids

• $\wedge$ and $\vee$ are idempotent: for all terms $a:L$, $a \wedge a = a$ and $a \vee a = a$

• for all $a:L$ and $b:L$, $a \wedge (a \vee b) = a$ and $a \vee (a \wedge b) = a$

A $\sigma$-complete lattice is a lattice $L$ with a function

such that

• for all $n:\mathbb{N}$ and $s:\mathbb{N} \to L$,

• for all $x:L$ and $s:\mathbb{N} \to L$, if $s(n) \wedge x = s(n)$ for all $n:\mathbb{N}$, then

A $\sigma$-frame is a $\sigma$-complete lattice with a function $\mathcal{t}:(L \times (\mathbb{N} \to L)) \to \mathbb{N} \to L$ such that

• for all $x:L$ and $s:\mathbb{N} \to L$, $\mathcal{t}(x,s)(n) = x \wedge s(n)$

• for all $x:L$ and $s:\mathbb{N} \to L$,

Sierpinski space

Sierpinski space $\Sigma$ is an initial $\sigma$-frame, and thus could be generated as a higher inductive type.