Dan Sommers writes:
> How would I "think of types as collections of their instances"?
The canonical example of a type as a collection of instances is an
enumeration, the simplest (useful) example of which is bool = {False,
True}.
In pre-big-integer Python, in principle you could do the same thing
with the collection of all 2^64-bit bit patterns, with addition
defined by a 2^64 x 2^64 tables with rows and columns indexed by the
bit patterns and the results in the table cells. So a type is a tuple
of a set and operations, which can also be defined as sets of the
above form. Floats would have the same underlying set and different
tables.
> For example, in what sense is the type list a collection of
> instances of lists?
This involves infinite sets, but in principle can be thought of in the
same way.
> In my mind, a type is a description/container/shortand for a
> collection of properties or behaviors of instances of that type.
That's one way to think about it, of course. But the strong
intuitions about the numeric tower (a natural number *is* an integer,
an integer *is* a real number, and so on) as well as some useful but
(intuitively) more artificial ideas such as a bool *is* a natural
number are just as well expressed as set inclusions. This is useful
in type theory, but explaining "how" is way beyond the scope of this
post (and this whole list, in fact).
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