# Qual è esattamente la differenza semantica tra set e type?

33

EDIT: ora ho fatto una domanda simile sulla differenza tra categorie e set.

Ogni volta che leggo della teoria dei tipi (che è certamente piuttosto informale), non riesco davvero a capire come si differenzia concretamente dalla teoria degli insiemi .

Capisco che esiste una differenza concettuale tra il dire "x appartiene a un set X" e "x è di tipo X", perché intuitivamente un set è solo una raccolta di oggetti, mentre un tipo ha determinate "proprietà". Tuttavia, gli insiemi sono spesso definiti anche in base alle proprietà, e se lo sono, allora ho difficoltà a capire in che modo questa distinzione conta in alcun modo.

Quindi, nel modo più concreto modo possibile, che cosa esattamente vuol implica circa $x$$x$ per dire che si tratta di tipo , rispetto a dire che si tratta di un elemento nel set di ?$T$$T$$S$$S$

(Puoi scegliere qualsiasi tipo e set che renda il confronto più chiaro).

Qual è il contesto in cui stai usando / ascoltando la parola "tipo"? È, come suggerisce il tuo nome, linguaggi di programmazione? Perché penso che le risposte che seguono assumano diversamente.
einpoklum - ripristina Monica il

@einpoklum, non sono sicuro al 100% su come descrivere quale sia il "contesto", ma sostanzialmente qualcosa di simile: sto cercando di capire il ruolo dei tipi in matematica. In sostanza, gli insiemi (come la vedo io) hanno due contesti: in primo luogo sono usati come raccolte di oggetti per fare matematica di tutti i giorni. In secondo luogo, sono oggetti nella teoria degli insiemi assiomatici, dove sono principalmente usati come uno strumento molto strano ma utile per parlare di matematica nella logica del primo ordine, lasciando che gli insiemi corrispondano a funzioni e numeri e così via. Sono principalmente interessato alla relazione tra "set" nel primo senso e "tipo".
user56834

Il ruolo di quali tipi? I tipi che vedi nei documenti di matematica / nei libri di testo o nei tipi di variabili nei programmi per computer?
einpoklum - ripristina Monica il

1
@einpoklum, questa domanda riguarda quelle nei documenti di matematica. (Anche se in realtà sono anche interessato a conoscere la differenza fondamentale tra tipi in matematica e tipi nei linguaggi di programmazione, se ce ne sono. Ma non era questo il problema).
user56834

Risposte:

29

Per capire la differenza tra insiemi e tipi, bisogna tornare a quelli idee pre- matematiche di "raccolta" e "costruzione", e vedere come insiemi e tipi li matematizzano .

Esiste una gamma di possibilità su ciò che riguarda la matematica. Due di questi sono:

1. Pensiamo alla matematica come un'attività in cui gli oggetti matematici sono costruiti secondo alcune regole (pensiamo alla geometria come l'attività di costruzione di punti, linee e cerchi con un righello e una bussola). Pertanto gli oggetti matematici sono organizzati in base al modo in cui sono costruiti e esistono diversi tipi di costruzione. Un oggetto matematico è sempre costruito in un modo unico, che determina il suo tipo unico.

2. Pensiamo alla matematica come a un vasto universo pieno di oggetti matematici preesistenti (pensiamo al piano geometrico come dato). Scopriamo, analizziamo e pensiamo a questi oggetti (osserviamo che ci sono punti, linee e cerchi nel piano). Li raccogliamo nel set . Solitamente raccogliamo elementi che hanno qualcosa in comune (ad esempio, tutte le linee che attraversano un determinato punto), ma in linea di principio un insieme può contenere una selezione arbitraria di oggetti. Un set è specificato dai suoi elementi e solo dai suoi elementi. Un oggetto matematico può appartenere a molti insiemi.

We are not saying that the above possibilities are the only two, or that any one of them completely describes what mathematics is. Nevertheless, each can view can serve as a useful starting point for a general mathematical theory that usefully describes a wide range of mathematical activities.

It is natural to take a type $T$$T$ and imagine the collection of all things that we can construct using the rules of $T$$T$. This is the extension of $T$$T$, and it is not $T$$T$ itself. For instance, here are two types that have different rules of construction, but they have the same extension:

1. The type of pairs $\left(n,p\right)$$(n, p)$ where $n$$n$ is constructed as a natural number, and $p$$p$ is constructed as a proof demonstrating that $n$$n$ is an even prime number larger than $3$$3$.

2. The type of pairs $\left(m,q\right)$$(m, q)$ where $m$$m$ is constructed as a natural number, and $q$$q$ is constructed as a proof demonstrating that $m$$m$ is an odd prime smaller than $2$$2$.

Yes, these are silly trivial examples, but the point stands: both types have nothing in their extension, but they have different rules of construction. In contrast, the sets

and
are equal because they have the same elements.

Note that type theory is not about syntax. It is a mathematical theory of constructions, just like set theory is a mathematical theory of collections. It just so happens that the usual presentations of type theory emphasize syntax, and consequently people end up thinking type theory is syntax. This is not the case. To confuse a mathematical object (construction) with a syntactic expression that represents it (a term former) is a basic category mistake that has puzzled logicians for a long time, but not anymore.

1
Beautiful, thank you! Could you clarify one detail? when you list the two types whose extension are both empty, you say that "the type whose elements are ...". Purely for my clarity, is this a 100% correct way of saying it? You said in the previous sentence that a type is not a collection, so it seems like it cannot have "elements" (which I associate with sets). Essentially, the way you've written it now, it is as if you're defining the Type according to the set that is its extention. If you didn't intend this, could you rephrase them more precisely to capture their idea as types?
user56834

The extension of a type is a very useful concept, and since it is a kind of collection, we can say "element of the extension of a type". This is cumbersome so it often gets abbreviated to just "element of a type". I removed the phrasing to decrease the possibility of confusion, but beware, it's common terminology.
Andrej Bauer

Thank you, this clarifies. So to follow up, is it correct to say the following? To say that an object is "of type T" means the same thing as, the object is "an element of the extension of T", so that there is a natural surjection from types to sets. But the converse does not hold, because any set can be constructed in multiple ways. Essentially, the difference between set and type is not important from the perspective of a particular object $x$$x$, in the sense that $x:T$$x:T$ and $x\in {X}_{T}$$x\in X_T$ (where ${X}_{T}$$X_T$ is the extension of $T$$T$) give us the exact same information about $x$$x$. However,
user56834

$x\in {X}_{T}$$x\in X_T$ rather than $x:T$$x:T$ does not tell us anything relevant about $x$$x$, but the same may not hold if we e.g. want to talk about superset-subset or type-subtype relations? Is that correct?
user56834

4
Yeah, one wonders where these books are. Someone should write them.
Andrej Bauer

11

To start, sets and types aren't even in the same arena. Sets are the objects of a first-order theory, such as ZFC set theory. While types are like overgrown sorts. To put it a different way, a set theory is a first-order theory within first-order logic. A type theory is an extension of logic itself. Martin-Löf Type Theory, for example, is not presented as a first-order theory within first-order logic. It's not that common to talk about sets and types at the same time.

As Discrete lizard states, types (and sorts) serve a syntactic function. A sort/type behaves as a syntactic category. It lets us know what expressions are well-formed. For a simple example using sorts, let's say we described the theory of vector spaces over an arbitrary field as a 2-sorted theory. We have a sort for scalars, $\mathsf{S}$$\mathsf S$, and a sort for vectors, $\mathsf{V}$$\mathsf V$. Among many other things, we'd have an operation for scaling: $\mathsf{s}\mathsf{c}\mathsf{a}\mathsf{l}\mathsf{e}:\mathsf{S}×\mathsf{V}\to \mathsf{V}$$\mathsf{scale}:\mathsf S\times \mathsf V\to \mathsf V$. This lets us know that $\mathsf{s}\mathsf{c}\mathsf{a}\mathsf{l}\mathsf{e}\left(\mathsf{s}\mathsf{c}\mathsf{a}\mathsf{l}\mathsf{e}\left(s,v\right),v\right)$$\mathsf{scale}(\mathsf{scale}(s,v),v)$ is simply not a well-formed term. In a type theoretic context, an expression like $f\left(x\right)$$f(x)$ requires $f$$f$ to have a type $X\to Y$$X\to Y$ for some types $X$$X$ and $Y$$Y$. If $f$$f$ does not have the type of a function, then $f\left(x\right)$$f(x)$ is simply not a well-formed expression. Whether an expression is of some sort or has some type is a meta-logical statement. It makes no sense to write something like: $\left(x:X\right)\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}y=3$$(x:X)\implies y=3$. First, $x:X$$x : X$ is simply not a formula, and second, it doesn't even conceptually make sense as sorts/types are what let us know which formulas are well-formed. We only consider the truth value of well-formed formulas, so by the time we're considering whether some formula holds, we better already know that it is well-formed!

In set theory, and particularly ZFC, the only non-logical symbol at all is the relation symbol for set membership, $\in$$\in$. So $x\in y$$x\in y$ is a well-formed formula with a truth value. There are no terms other than variables. All the usual notation of set theory is a definitional extension to this. For example, a formula like $f\left(x\right)=y$$f(x)=y$ is often taken to be shorthand for $\left(x,y\right)\in f$$(x,y)\in f$ which itself may be taken as shorthand for $\mathrm{\exists }p.p\in f\wedge p=\left(x,y\right)$$\exists p.p\in f\land p=(x,y)$ which is shorthand for

$\mathrm{\exists }p.p\in f\wedge \left(\mathrm{\forall }z.z\in p\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\left[z=x\vee \left(\mathrm{\forall }w.w\in z⇔w=y\right)\right]\right)$
At any rate, any set can take the place of $f$$f$ and everything is a set! As I pointed out in a different question recently, $\pi \left(7\right)=3$$\pi(7)=3$ where $\pi$$\pi$ is the real number is a completely legitimate and meaningful (and potentially even true) set theoretic expression. Basically, anything you write that parses in set theory can be given some meaning. It may be a completely spurious meaning, but it has one. Sets are also "first-class" objects in set theory. (They better be as they are the only objects usually.) A function like
is a completely legitimate function in set theory. There is nothing even remotely analogous to this in type theory. The closest would be to use codes for a Tarskian universe. Sets are the objects of set theory; types are not the object of type theory.

A type is not a collection of things (neither is a set for that matter...), and it is not defined by a property. A type is a syntactic category that lets you know what operations are applicable to terms of that type and which expressions are well-formed. From a propositions-as-types perspective, what types are classifying are the valid proofs of the proposition to which the type corresponds. That is, the well-formed (i.e. well-typed) terms of a given type correspond to the valid proofs (which are also syntactic objects) of the corresponding proposition. Nothing like this is happening in set theory.

Set theory and type theory are really not anything alike.

1
It is false that types are only syntactic entities.
Andrej Bauer

1
This is very helpful, but one main point in your answer bugs me. It seems to me that it is a mistake (which many people make, or alternatively it is not a mistake and I'm wrong), to say that "a set is not a collection of things". I would say that a set IS a collection of things. That is the most basic essential property of a set. In fact, how would we possibly know that e.g. ZFC are the right axioms to choose (rather than completely arbitrary formulas), without being able to tell that they are true given that sets are collections of objects? Of course, I understand that ...
user56834

Axiomatic set theory treats sets as objects, and $\in$$\in$ as just a symbol, because axiomatic set theory is not a mathematical structure in the mathematical logic sense.
user56834

1
@Programmer2134 To answer that, we would have to get into the semantic meaning of the word "collection." We can't be sure they are "right" unless you take the time to precisely define what "right" means. However, what we can say is that "set" is the result of over a hundred years of mathematicians beating on the concept of a collection, seeking a consistent system which matches the intuitive concept of a collection. To achieve that consistency, they had to make decisions. For example, sets aren't the only collection in mathematics. A "class" also describes a collection.
Cort Ammon - Reinstate Monica

1
@AndrejBauer I'm taking a (mostly) non-philosophical stance and not trying to explain what types "really" are but more how they're used. (I say "serves as" and "behaves as" at the beginning but I did slip into an "is" at the end.) There is a risk of thinking that a variable $x$$x$ being of type $T$$T$ means that the only "values" that $x$$x$ can "take" are the (presumably closed) terms of type $T$$T$. This isn't true and is not implied by anything I say above. I agree that you can see types as more than syntactic entities, but I think the different syntactic role types play makes a clear contrast to sets.
Derek Elkins

9

In practice, claiming that $x$$x$ being of type $T$$T$ usually is used to describe syntax, while claiming that $x$$x$ is in set $S$$S$ is usually used to indicate a semantic property. I will give some examples to clarify this difference in usage of types and sets. For the difference in what types and sets actually are, I refer to Andrej Bauer's answer.

## An example

To clarify this distinction, I will use the example given in Herman Geuvers' lecture notes. First, we look at an example of inhabiting a type:

$3+\left(7\ast 8{\right)}^{5}:\mathrm{N}\mathrm{a}\mathrm{t},$
and an example of being member of a set:
$3\in \left\{n\in \mathbb{N}\mid \mathrm{\forall }x,y,z\in {\mathbb{N}}^{+}\left({x}^{n}+{y}^{n}\ne {z}^{n}\right)\right\}$

The main difference here is that to test whether the first expression is a natural number, we don't have to compute some semantic meaning, we merely have to 'read off' the fact that all literals are of type Nat and that all operators are closed on the type Nat.

However, for the second example of the set, we have to determine the semantic meaning of the $3$$3$ in the context of the set. For this particular set, this is quite hard: the membership of $3$$3$ for this set is equivalent to proving Fermat's last theorem! Do note that, as stated in the notes, the distinction between syntax and semantics cannot always be drawn that clearly. (and you might even argue that even this example is unclear, as Programmer2134 mentions in the comments)

## Algorithms vs Proofs

To summarize, types are often used for 'simple' claims on the syntax of some expression, such that membership of a type can be checked by an algorithm, while to test membership of a set, we would in usually require a proof.

To see why this distinction is useful, consider a compiler of a typed programming language. If this compiler has to create a formal proof to 'check types', the compiler is asked to do an almost impossible task (automated theorem proving is, in general, hard). If on the other hand the compiler can simply run an (efficient) algorithm to check the types, then it can realistically perform the task.

## A motivation for a strict(er) interpretation

There are multiple interpretations of the semantic meaning of sets and types. While under the distinction made here extensional types and types with undecidable type-checking (such as those used in NuPRL, as mentioned in the comments) would not be 'types', others are of course free to call them as such (just as free as they are as to call them something else, as long as their definitions fit).

However, we (Herman Geuvers and I), prefer to not throw this interpretation out of the window, for which I (not Herman, although he might agree) have the following motivation:

First of all, the intention of this interpretation isn't that far from that of Andrej Bauer. The intention of a syntax is usually to describe how to construct something and having an algorithm to actually construct it is generally useful. Furthermore, the features of a set are usually only needed when we want a semantic description, for which undecidability is allowed.

So, the advantage of our more stricter description is to keep the separation simpler, to get a distinction more directly related to common practical usage. This works well, as long as you don't need or want to loosen your usage, as you would for, e.g. NuPRL.

3
Type checking doesn't need to be decidable (though it is certainly desirable). NuPRL, for instance, does require the user to provide a proof that a term inhabits a type.
Derek Elkins

3
Thank you. Things are becoming more clear. Heres what I'm wondering still: isn't there a semantic element to type theory and a syntactic element to set theory? For example, we can instead of seeing your "$3\in ...$$3\in ...$" statement as a semantic statement, see it as a proposition in axiomatic set theory, no? Also, doesn't the type "Nat" have a semantic meaning, namely that whatever precedes it is a natural number? So it still confuses me to say rhat sets are semantic and types syntactic properties.
user56834

1
@DerekElkins I'm not familiar with NuPRL, but e.g. the proof assistant Coq most certainly does type checking by itself (i.e, is the provided term of the 'type of my theorem'). How does NuPRL verify the proof if the user has to 'prove' the fact that a term of a certain type? (in other words, this sounds like NuPRL doesn't use the Curry-Howard correspondence, so what does it use?)
Discrete lizard

1
@Discretelizard I'm not saying NuPRL is typical. It is definitely the usual case for type checking to be decidable. I highly recommend becoming familiar with it just because it takes a fairly different path. NuPRL is a Curry-style rather than Church-style calculus which makes it more of a type refinement system. At any rate, instead of just writing terms (or tactics that produce terms), you have essentially an LCF-style proof system for typing derivations themselves. Arguably, the derivations are what's important, and it's a bit of a "fluke" that we can infer them from terms.
Derek Elkins

3
I elaborated in my answer why it is detrimental to think of type theory as "syntactic". Your first sentence is the offeding one, when you say that "inhabiting a type is a syntactic property". There's as much truth in that as in saying that "being an element of a set is to stand on the left of the symbol $\in$$\in$".
Andrej Bauer

4

I believe that one of the most concrete differences about sets and types is the difference in the way the "things" in your mind are encoded into the formal language.

Both sets and types allow you to speak about things, and collections of things. The main difference is that with sets, you can ask any question you want about things and it will maybe be true, maybe not; while with types, you first have to prove that the question makes sense.

For example, if you have booleans $\mathbb{B}=\left\{\mathrm{true},\mathrm{false}\right\}$$\mathbb B=\{\operatorname{true},\operatorname{false}\}$ and natural numbers $\mathbb{N}=\left\{0,1,\dots \right\}$$\mathbb N=\{0, 1,\dots \}$, with types, you can not ask if $\mathrm{true}=1$$\operatorname{true}=1$ which you can with sets.

One way to interpret this is that with sets, everything is encoded into a single collection: the collection of all sets. $0$$0$ is encoded as $\left[0\right]=\left\{\right\}$$[0]=\{\}$, $n+1$$n+1$ is encoded as $\left[n+1\right]=\left\{\left[n\right]\right\}\cup \left[n\right]$$[n+1]=\{[n]\}\cup [n]$ and $\mathrm{true}$$\operatorname{true}$ and $\mathrm{false}$$\operatorname{false}$ can be encoded by any two distinct sets. So that it actually makes sense to ask if $\mathrm{true}=1$$\operatorname{true}=1$, as it can be understood as asking if "the encoding chosen for $\mathrm{true}$$\operatorname{true}$ is the same as the encoding chosen for $1$$1$". But the answer to this question could change if we chose another encoding: it is about the encodings and not really about the things.

You can then think of types as describing the encoding of the things inside it. With types, to ask the question of whether $a=b$$a=b$, you first have to show that $a$$a$ and $b$$b$ have the same type, i.e. that they were encoded in the same way, which prohibits questions such as $\mathrm{true}=1$$\operatorname{true}=1$. You could still want to have a big type $S$$S$ in which both $\mathbb{B}$$\mathbb B$ and $\mathbb{N}$$\mathbb N$ could be encoded, and then given two encodings ${\iota }_{\mathbb{B}}:\mathbb{B}\to S$$\iota_\mathbb B:\mathbb B\to S$ and ${\iota }_{\mathbb{N}}:\mathbb{N}\to S$$\iota_\mathbb N:\mathbb N\to S$, you could ask whether ${\iota }_{\mathbb{B}}\left(\mathrm{true}\right)={\iota }_{\mathbb{N}}\left(1\right)$$\iota_\mathbb B(\operatorname{true})=\iota_\mathbb N(1)$ but the fact that this question depends on the encodings (and the choice of encodings) is now explicit.

Note that in those cases, whether the question made sense was actually easy to see but it could be much harder as in, for example, $\left(\mathrm{if}\text{very_hard_question}\mathrm{then}1\mathrm{else}\mathrm{true}\right)=1$$(\operatorname{if} \text{very_hard_question} \operatorname{then} 1 \operatorname{else} \operatorname{true})=1$.

In summary, sets let you ask any question you want, but types force you to make encodings explicit when the answer may depend on them.

You are probably thinking of one specific kind of set theory (something along a single-sorted theory a la ZFC). However, there are other kinds of set theory which require lots of checking that it makes sense. And the way set theory is used in practice is much closer to these other set theories. Do you think a student could ask "Is $\mathbb{R}$$\mathbb{R}$ an element of $\mathrm{sin}\left(\sqrt{2}\right)$$\sin(\sqrt{2})$?" without being scolded? The distinction between type theory and set theory is not in the formalism, it's in the meaning.
Andrej Bauer

@AndrejBauer Right. Would you agree that this answer gives a differences between single-sorted theories (including most set theories, or at least the most common ones), and multi-sorted theories (including all (?) type theories)?
xavierm02

Even in a single-sorted theory you have to distinguish terms from formulas...
Andrej Bauer

@AndrejBauer I don't understand your second comment.
xavierm02

A single-sorted first-order theory has two syntactic categories: logical formulas and terms. One has to make sure they're not mixed up, or else you could end up writing "$\mathrm{\varnothing }\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\left(\mathrm{\forall }x\in X.\varphi \left(x\right)\right)\in \mathbb{N}$$\emptyset \implies (\forall x \in X . \phi(x)) \in \mathbb{N}$".
Andrej Bauer
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