# nLab coherent mathematics

foundations

## Removing axioms

#### Constructivism, Realizability, Computability

intuitionistic mathematics

# Contents

## Idea

Coherent mathematics is mathematics done using only coherent logic.

Examples of coherent mathematics include mathematics done in an arithmetic pretopos, which can be defined using only coherent logic, as well as mathematics done in any dependent type theory which does not have function types nor dependent function types, such as the internal type theory of an arithmetic pretopos as a fragment of geometric type theory.

Since coherent logic does not have a negation operator, it is manifestly constructive, since one cannot even express the law of excluded middle or double negation. Similarly, coherent mathematics is manifestly strongly predicative, since having function sets in the theory implies having implication and $\Delta_0$-universal quantifiers, as every cartesian closed coherent category is a Heyting category.

## Equality and apartness

Since coherent mathematics does not have negation, one cannot use negation to define denial inequality as the negation of equality. In particular, what fails is that the definition of denial inequality involves a double implication

$\big( (a = b) \implies \bot \big) \implies (a \neq b)$

which is not allowed in coherent mathematics. On the level of coherent sequents, single implications $\phi \implies \psi$ are expressed as entailment, the sequent $\phi \vdash \psi$. But double implications $(\phi \implies \psi) \implies \varphi$ become the sequent $(\phi \implies \psi) \vdash \varphi$, which cannot be simplified down further to get rid of the remaining implication. Instead, one has to use the positive notion of apartness relation $a \# b$ to express the notion of “not being equal”.

Similarly to the fact that one cannot define denial inequality in coherent mathematics, one cannot define a tight apartness relation, since the definition involves negation of an apartness relation implying equality, which is a double implication

$((a \# b) \implies \bot) \implies (a = b)$

As a result, concepts such as Heyting fields which are important in constructive mathematics cannot be defined, and one has to use alternative concepts such as local rings instead.

Other things have to be expressed positively as well - for example, in the definition of a local ring, one cannot say that the non-invertible elements form an ideal $I \subseteq A$, since that would involve an implication

$((\exists y \in R.x \cdot y = 1) \implies \bot) \vdash x \in I$

or in the structural case with injection $i:I \hookrightarrow R$,

$((\exists y \in R.x \cdot y = 1) \implies \bot) \vdash \exists z \in I.i(z) = x$

Instead, one has to say that the invertible elements form an anti-ideal $A \subseteq R$.

$\exists y \in R.x \cdot y = 1 \vdash x \in A \quad \mathrm{material}$
$\exists y \in R.x \cdot y = 1 \vdash \exists z \in A.i(z) = x \quad \mathrm{structural}$

Furthermore, in the definition of an anti-ideal itself, when one says that $0$ is not in the anti-ideal, one has to express it as a sequent $0 \in A \vdash \bot$, or in the structural case with injection $i:A \hookrightarrow R$, that $\exists x \in A.i(x) = 0 \vdash \bot$. Alternatively, if the set has an apartness relation one could just say that every element in the ideal is apart from zero, $0 \# x$.

The degree of a polynomial is only well-defined in any ring with an apartness relation. Anything that depends on well-defined degrees of a polynomial, such as algebraic closure, also is only well-defined for a ring with an apartness relation.

### Decidable equality

Sets with decidable equality are however still definable in coherent mathematics; they are sets $S$ equipped with an apartness relation such that for all elements $a \in S$ and $b \in S$, $a = b$ or $a \# b$. This is expressed by the sequent

$a \in S, b \in S \vdash (a = b) \vee (a \# b)$

This means that one could work in predicative classical mathematics (i.e. a Boolean category) by stipulating that every set has decidable equality.

Nonetheless, even without assuming the above, a lot of algebraic number theory is coherent because the rings and number fields studied have decidable equality.

## Predicates, subsets, and restricted separation

The axiom schema of restricted separation in the set theory axioms states that for each set $S$ and each predicate $x \in S \vdash P(x)$, one could construct

• a set $\{x \in S \vert P(x)\}$
• a function $i:\{x \in S \vert P(x)\} \to S$
• a predicate $y \in \{x \in S \vert P(x)\}, z \in \{x \in S \vert P(x)\}, i(y) = i(z) \vdash y = z$
• a predicate $x \in R, \exists y \in \{x \in S \vert P(x)\}.x = i(y) \vdash P(x)$,
• a predicate $x \in R, P(x) \vdash \exists y \in \{x \in S \vert P(x)\}.x = i(y)$

This implies that $\{x \in S \vert P(x)\}$ is a subset of $S$ in the structural sense.

Assuming that the ambient logic is coherent logic, restricted separation automatically implies that the sets and functions form a coherent category, since the poset of subsets of each set being a distributive lattice follows from the rules for coherent logic and restricted separation.