1.1 Type theory versus set theory

We note that a set-theoretic foundation has two “layers”: the deductive system of first-order logic, and, formulated inside this system, the axioms of a particular theory, such as ZFC. Thus, set theory is not only about sets, but rather about the interplay between sets (the objects of the second layer) and propositions   (the objects of the first layer).

By contrast, type theory is its own deductive system: it need not be formulated inside any superstructure, such as first-order logic. Instead of the two basic notions of set theory, sets and propositions, type theory has one basic notion: types. Propositions (statements which we can prove, disprove, assume, negate, and so on11Confusingly, it is also a common practice (dating back to Euclid) to use the word “proposition” synonymously with “theorem”. We will confine ourselves to the logician’s usage, according to which a proposition is a statement susceptible to proof, whereas a theorem (or “lemma” or “corollary”) is such a statement that has been proven. Thus “$0=1$” and its negation  $\neg(0=1)$” are both propositions, but only the latter is a theorem.) are identified with particular types, via the correspondence shown in Table 1 (http://planetmath.org/constructivity#S0.T1). Thus, the mathematical activity of proving a theorem is identified with a special case of the mathematical activity of constructing an object—in this case, an inhabitant of a type that represents a proposition.

This leads us to another difference between type theory and set theory, but to explain it we must say a little about deductive systems in general. Informally, a deductive system is a collection  of rules for deriving things called judgments. If we think of a deductive system as a formal game, then the judgments are the “positions” in the game which we reach by following the game rules. We can also think of a deductive system as a sort of algebraic theory, in which case the judgments are the elements (like the elements of a group) and the deductive rules are the operations  (like the group multiplication  ). From a logical point of view, the judgments can be considered to be the “external” statements, living in the metatheory  , as opposed to the “internal” statements of the theory itself.

In the deductive system of first-order logic (on which set theory is based), there is only one kind of judgment: that a given proposition has a proof. That is, each proposition $A$ gives rise to a judgment “$A$ has a proof”, and all judgments are of this form. A rule of first-order logic such as “from $A$ and $B$ infer $A\wedge B$” is actually a rule of “proof construction” which says that given the judgments “$A$ has a proof” and “$B$ has a proof”, we may deduce that “$A\wedge B$ has a proof”. Note that the judgment “$A$ has a proof” exists at a different level from the proposition $A$ itself, which is an internal statement of the theory.

The basic judgment of type theory, analogous to “$A$ has a proof”, is written “$a:A$” and pronounced as “the term $a$ has type $A$”, or more loosely “$a$ is an element of $A$” (or, in homotopy type theory, “$a$ is a point of $A$”). When $A$ is a type representing a proposition, then $a$ may be called a witness to the provability of $A$, or evidence of the truth of $A$ (or even a proof of $A$, but we will try to avoid this confusing terminology). In this case, the judgment $a:A$ is derivable  in type theory (for some $a$) precisely when the analogous judgment “$A$ has a proof” is derivable in first-order logic (modulo differences in the axioms assumed and in the encoding of mathematics, as we will discuss throughout the book).

On the other hand, if the type $A$ is being treated more like a set than like a proposition (although as we will see, the distinction can become blurry), then “$a:A$” may be regarded as analogous to the set-theoretic statement “$a\in A$”. However, there is an essential difference in that “$a:A$” is a judgment whereas “$a\in A$” is a proposition. In particular, when working internally in type theory, we cannot make statements such as “if $a:A$ then it is not the case that $b:B$”, nor can we “disprove” the judgment “$a:A$”.

A good way to think about this is that in set theory, “membership” is a relation   which may or may not hold between two pre-existing objects “$a$” and “$A$”, while in type theory we cannot talk about an element “$a$” in isolation: every element by its very nature is an element of some type, and that type is (generally speaking) uniquely determined. Thus, when we say informally “let $x$ be a natural number  ”, in set theory this is shorthand for “let $x$ be a thing and assume that $x\in\mathbb{N}$”, whereas in type theory “let $x:\mathbb{N}$” is an atomic statement: we cannot introduce a variable without specifying its type.

At first glance, this may seem an uncomfortable restriction  , but it is arguably closer to the intuitive mathematical meaning of “let $x$ be a natural number”. In practice, it seems that whenever we actually need$a\in A$” to be a proposition rather than a judgment, there is always an ambient set $B$ of which $a$ is known to be an element and $A$ is known to be a subset. This situation is also easy to represent in type theory, by taking $a$ to be an element of the type $B$, and $A$ to be a predicate  on $B$; see §3.5 (http://planetmath.org/35subsetsandpropositionalresizing).

A last difference between type theory and set theory is the treatment of equality. The familiar notion of equality in mathematics is a proposition: e.g. we can disprove an equality or assume an equality as a hypothesis   . Since in type theory, propositions are types, this means that equality is a type: for elements $a,b:A$ (that is, both $a:A$ and $b:A$) we have a type “$a=_{A}b$”. (In homotopy type theory, of course, this equality proposition can behave in unfamiliar ways: see §1.12 (http://planetmath.org/112identitytypes), and the rest of the book). When $a=_{A}b$ is inhabited, we say that $a$ and $b$ are (propositionally) equal.

However, in type theory there is also a need for an equality judgment, existing at the same level as the judgment “$x:A$”. This is called judgmental equality or definitional equality, equality, definitionaland we write it as $a\equiv b:A$ or simply $a\equiv b$. It is helpful to think of this as meaning “equal by definition”. For instance, if we define a function $f:\mathbb{N}\to\mathbb{N}$ by the equation $f(x)=x^{2}$, then the expression $f(3)$ is equal to $3^{2}$ by definition. Inside the theory, it does not make sense to negate or assume an equality-by-definition; we cannot say “if $x$ is equal to $y$ by definition, then $z$ is not equal to $w$ by definition”. Whether or not two expressions are equal by definition is just a matter of expanding out the definitions; in particular, it is algorithmically decidable (though the algorithm  is necessarily meta-theoretic, not internal to the theory).

As type theory becomes more complicated, judgmental equality can get more subtle than this, but it is a good intuition to start from. Alternatively, if we regard a deductive system as an algebraic theory, then judgmental equality is simply the equality in that theory, analogous to the equality between elements of a group—the only potential for confusion is that there is also an object inside the deductive system of type theory (namely the type “$a=b$”) which behaves internally as a notion of “equality”.

The reason we want a judgmental notion of equality is so that it can control the other form of judgment, “$a:A$”. For instance, suppose we have given a proof that $3^{2}=9$, i.e. we have derived the judgment $p:(3^{2}=9)$ for some $p$. Then the same witness $p$ ought to count as a proof that $f(3)=9$, since $f(3)$ is $3^{2}$ by definition. The best way to represent this is with a rule saying that given the judgments $a:A$ and $A\equiv B$, we may derive the judgment $a:B$.

Thus, for us, type theory will be a deductive system based on two forms of judgment:

Judgment Meaning
$a:A$ $a$ is an object of type $A$
$a\equiv b:A$ $a$ and $b$ are definitionally equal objects of type $A$

When introducing a definitional equality, i.e., defining one thing to be equal to another, we will use the symbol “$܃\!\!\equiv$”. Thus, the above definition of the function $f$ would be written as $f(x)܃\!\!\equiv x^{2}$.

Because judgments cannot be put together into more complicated statements, the symbols “$:$” and “$\equiv$” bind more loosely than anything else.22In formalized type theory, commas and turnstiles can bind even more loosely. For instance, $x:A,y:B\vdash c:C$ is parsed as $((x:A),(y:B))\vdash(c:C)$. However, in this book we refrain from such notation until \autorefcha:rules. Thus, for instance, “$p:x=y$” should be parsed as “$p:(x=y)$”, which makes sense since “$x=y$” is a type, and not as “$(p:x)=y$”, which is senseless since “$p:x$” is a judgment and cannot be equal to anything. Similarly, “$A\equiv x=y$” can only be parsed as “$A\equiv(x=y)$”, although in extreme cases such as this, one ought to add parentheses anyway to aid reading comprehension. Moreover, later on we will fall into the common notation of chaining together equalities — e.g. writing $a=b=c=d$ to mean “$a=b$ and $b=c$ and $c=d$, hence $a=d$” — and we will also include judgmental equalities in such chains. Context usually suffices to make the intent clear.

This is perhaps also an appropriate place to mention that the common mathematical notation “$f:A\to B$”, expressing the fact that $f$ is a function from $A$ to $B$, can be regarded as a typing judgment, since we use “$A\to B$” as notation for the type of functions from $A$ to $B$ (as is standard practice in type theory; see §1.4 (http://planetmath.org/14dependentfunctiontypes)).

Judgments may depend on assumptions of the form $x:A$, where $x$ is a variable and $A$ is a type. For example, we may construct an object $m+n:\mathbb{N}$ under the assumptions that $m,n:\mathbb{N}$. Another example is that assuming $A$ is a type, $x,y:A$, and $p:x=_{A}y$, we may construct an element $p^{-1}:y=_{A}x$. The collection of all such assumptions is called the context; from a topological point of view it may be thought of as a “parameter space”. In fact, technically the context must be an ordered list of assumptions, since later assumptions may depend on previous ones: the assumption $x:A$ can only be made after the assumptions of any variables appearing in the type $A$.

If the type $A$ in an assumption $x:A$ represents a proposition, then the assumption is a type-theoretic version of a hypothesis: we assume that the proposition $A$ holds. When types are regarded as propositions, we may omit the names of their proofs. Thus, in the second example above we may instead say that assuming $x=_{A}y$, we can prove $y=_{A}x$. However, since we are doing “proof-relevant” mathematics, we will frequently refer back to proofs as objects. In the example above, for instance, we may want to establish that $p^{-1}$ together with the proofs of transitivity and reflexivity  behave like a groupoid    ; see http://planetmath.org/node/87569Chapter 2.

Note that under this meaning of the word assumption, we can assume a propositional equality (by assuming a variable $p:x=y$), but we cannot assume a judgmental equality $x\equiv y$, since it is not a type that can have an element. However, we can do something else which looks kind of like assuming a judgmental equality: if we have a type or an element which involves a variable $x:A$, then we can substitute any particular element $a:A$ for $x$ to obtain a more specific type or element. We will sometimes use language like “now assume $x\equiv a$” to refer to this process of substitution, even though it is not an assumption in the technical sense introduced above.

By the same token, we cannot prove a judgmental equality either, since it is not a type in which we can exhibit a witness. Nevertheless, we will sometimes state judgmental equalities as part of a theorem, e.g. “there exists $f:A\to B$ such that $f(x)\equiv y$”. This should be regarded as the making of two separate judgments: first we make the judgment $f:A\to B$ for some element $f$, then we make the additional judgment that $f(x)\equiv y$.

In the rest of this chapter, we attempt to give an informal presentation  of type theory, sufficient for the purposes of this book; we give a more formal account in http://planetmath.org/node/87881Appendix A. Aside from some fairly obvious rules (such as the fact that judgmentally equal things can always be substituted for each other), the rules of type theory can be grouped into type formers. Each type former consists of a way to construct types (possibly making use of previously constructed types), together with rules for the construction and behavior of elements of that type. In most cases, these rules follow a fairly predictable pattern, but we will not attempt to make this precise here; see however the beginning of §1.5 (http://planetmath.org/15producttypes) and also http://planetmath.org/node/87578Chapter 5.

An important aspect of the type theory presented in this chapter is that it consists entirely of rules, without any axioms. In the description of deductive systems in terms of judgments, the rules are what allow us to conclude one judgment from a collection of others, while the axioms are the judgments we are given at the outset. If we think of a deductive system as a formal game, then the rules are the rules of the game, while the axioms are the starting position. And if we think of a deductive system as an algebraic theory, then the rules are the operations of the theory, while the axioms are the generators for some particular free model of that theory.

In set theory, the only rules are the rules of first-order logic (such as the rule allowing us to deduce “$A\wedge B$ has a proof” from “$A$ has a proof” and “$B$ has a proof”): all the information about the behavior of sets is contained in the axioms. By contrast, in type theory, it is usually the rules which contain all the information, with no axioms being necessary. For instance, in §1.05 (http://planetmath.org/15producttypes) we will see that there is a rule allowing us to deduce the judgment “$(a,b):A\times B$” from “$a:A$” and “$b:B$”, whereas in set theory the analogous statement would be (a consequence of) the pairing axiom.

The advantage of formulating type theory using only rules is that rules are “procedural”. In particular, this property is what makes possible (though it does not automatically ensure) the good computational properties of type theory, such as “canonicity”. However, while this style works for traditional type theories, we do not yet understand how to formulate everything we need for homotopy type theory in this way. In particular, in §2.9 (http://planetmath.org/29pitypesandthefunctionextensionalityaxiom),§2.10 (http://planetmath.org/210universesandtheunivalenceaxiom),http://planetmath.org/node/87579Chapter 6 we will have to augment the rules of type theory presented in this chapter by introducing additional axioms, notably the univalence axiom. In this chapter, however, we confine ourselves to a traditional rule-based type theory.

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