The theory of automata, the digital, all-or-none type as discussed up to now, is certainly a chapter in formal logic. [...] The set on the right is just a representation of a particular counting number that might appear to be in no particular need of elaboration. [...] As mentioned already, one is often interested in a particular type of element such as natural numbers in computability or the reals in analysis. We could assume the reals R are simply given to us, and we can start forming set of reals right away. [...] In typical applications, a handful of rounds is enough to obtain the sets we need. [...] Extensionality has another important side-effect: it can be very difficult to establish equality even when one of the sets in question is trivial. [...] We can stretch the list description a little bit and omit some of the elements if they are obvious from context. So we write A = {a1, a2, . . . [...] Take this with a grain of salt, for us this is just a way to motivate some of the material. [...] In order to have a general complementation operation, we need to fix some set U as the universe of discourse, and only consider sets A with elements from U: we can then set A− = U − A. [...] A good starting point for this project is to find a way to implement lists7. Since sets are unordered and do not allow for repetitions, we cannot simply use the set of the list elements, not even for pairs, lists of length 2. [...] The question then is what the elements of Nn should be, and a reasonable answer is that they should be all the smaller numerals: Nn = {N0, N1, . . . , Nn−1}.
- Pages
- 333
- Published in
- United States of America
