OK, I think
--- paul.hudak@yale.edu wrote: Jerzy Karczmarczuk wrote: that this subject matured enough to rest in peace...
I would have to
agree with that, although... Since the subject is not going to rest, why not also jump in?
Well, each partial list is finite.
But the
elements comprising
I think quite a few people would agree that a finite list is one ending in []. So 1:_|_ is a partial list, but not a finite one. 1:[] is a finite list. limit of a chain IS the maximal element of the set of all the chain, since the LUB, in the case of a chain, is
unique, and thus we don't have to worry about choosing the "least" element (i.e. it reduces to the upper bound, or maximal element).
That doesn't quite make sense, IMHO. The chain _|_, 1:_|_, 1:1:_|_, 1:1:1:_|_, ... has the limit (or upper bound) "ones", but "ones" does not appear in the chain. An upper bound of a set may not appear in the set. But the maximal element of a set by definition ("element of") necessarily is in the set. So you cannot use the two notions interchangably. BTW, if the limit of a chain would always appear in the chain (by virtue of being the maximal element of the set of all elements comprising the chain), then every chain would be stationary eventually. That would quite simplify denotational semantics, wouldn't it?
So I'd say that Brian has at least come close to discovering God :-)
And come to the conclusion, in a later mail, that both: A. "the list [0,1,2,3,4,5,..] is longer than [1,2,3,4,5,..]" and B. "ie [0,1,2,3,4,..] is the same length as [1,2,3,4,5,..]" I think not. Obviously not. I know that this conclusion was qualified by "assuming that they all grow at the same rate", which of course has no counterpart in the denotational semantics setting discussed above. So it's not a wrong statement, just one that cannot really be formulated. Well, anyway, didn't want to be nitpicking. But I think it's important that if we want to think about Haskell's semantics denotationally (which in itself is somewhat problematic), we should at least be careful not to blur concepts. And asserting that the limit of a chain is always an element of that chain is a dangerous oversimplification that does not work out. Ciao, Janis Voigtlaender.