| >They're not, in general, subsets of some larger universe of values You have a Computer. The Computer has N bytes of memory. You want to declare a Boolean type variable. You tell your compiler this by coding it in the language of your choice. The compiler will typically allocate 1 to 8 bytes of memory to store that Boolean value -- depending on such things as how many bits your CPU is, what compiler options are, if variables should be byte or dword or qword aligned in memory, etc. >They're not, in general, subsets of some larger universe of values But the thing is, that memory allocated for the Boolean variable is now constrained -- to be a subset of all of the previously permissible bit patterns for that memory. It is constrained to be either 0 (representing False) or 1, (representing True). If that Boolean value is 8 bits long (let's say) -- then the only two permissible possibilties that exist for it -- are either 00000000 (0 - False) or 00000001 (1 - True). >They're not, in general, subsets of some larger universe of values That type, the Boolean -- is very much a subset -- of a larger universe of values... For a given Byte it can also be determined if it belongs to the Boolean Type
-- via a simple function (aka "series of steps", aka algorithm). Basically, just compare that Byte's value to 0 or 1. If it's either a 0 or a 1, then it's a member -- and if it isn't one of these values then it isn't. That small series of steps -- is a function. A function which determines membership of given data -- an input (in this case, a byte) -- to a type (in this case, a Boolean type). To validate or repudiate type membership (or lack thereof) of something more complex -- a more complex function/algorithm/series of steps -- may be needed -- but the point is, that's how it's done... >They're not, in general, subsets of some larger universe of values So they are all-inclusive sets of some larger universe of values? ? If that's the case -- then why do neither computer programs nor mathematical constructs that use types -- use a single solitary type that represents the larger universe of all possible values everywhere that a type is used? ? |
You're confusing several things here.
- a term is not its runtime representation
- a type is not the set of all terms that inhabit it (though you can get away with pretending it is in simple cases)
- and more generally, operational semantics are not denotational semantics
> So they are all-inclusive sets of some larger universe of values?
No, they're simply not sets. Types are primitive notions in type theories, the same way that sets are the primitive notions of set theory. No reduction of one to the other is necessary or, typically, desirable.
And even if you try to build type theory on set-theoretic foundations - which is 100% the wrong choice if you want to apply it to computational problems down the line - you're still going to run into problems once things start getting recursive. Consider, for example, the type of "hyperfunctions" given by
`type Hyper a b = Hyper b a -> b`
This is too big to be a set, for the usual self-containment reasons. But it's a perfectly legitimate Haskell type, modulo syntax. I've used it in real code.
> use a single solitary type that represents the larger universe of all possible values everywhere that a type is used?
That's what a dynamically typed programming language is.