This is a selection from messages that Dr Bill Richter has posted to comp.lang.scheme. ================================================================ 7 June 2004: If you don't know what Hausdorff means, let's just say this: the real line R is Hausdorff, and non-Hausdorff is really strange. ================================================================ 21 June 2004: That's hot, Joe! My favorite set theory book, by Just & Weese, says that prior to Goedel incompleteness, no one had ever proved a theorem in Number Theory that independent of the Peano axioms. And of course, the Goedel sentence is pretty weird, not clearly related to any Number Theory we know or care about. This sounds a lot more interesting. Can you say more? I couldn't grok your code. Where's the kerosene anyway? My point is that you can do great things with Math, like construct DS function with or without Scott models, but only because Math has such strong (set theory) axioms. But we know from Goedel incompleteness that are axioms aren't complete. Your turn! ================================================================ 3 July 2004: ....But I'm using much the same ZFC axioms that you need for CPO/Scott approach. Perhaps I'm not using the power set axiom! Scott's P(omega) is the power set of the natural numbers. I doubt you're ready to prove that the compositionality is dependent on the ZFC power set axiom! ================================================================ 3 July 2004: F-F and I are using a very complex mechanism: the ZFC axioms! It's the same ZFC axioms you need (except perhaps the power set axiom) for the CPO/Scott approach. My Math biz bias tells me it's a deficiency to learn the CPO/Scott approach without going through the ZFC axiom biz which makes it run. I could be wrong, but I thought Daniel said he was a constructionist who (earlier) liked the CPO/Scott approach. I apologize if I got Daniel wrong, but that's Will's "tremendous confusion". If you're using Scott models, you're no constructionist, you're a ZFC-er. ================================================================ 23 July 2004: A Texan prof at NWU actually studied in R L Moore's school, and he said that R L stood for Robert Lee, and that this was a common name, at least back then, for Confederacy-tolerant reasons. Will, if I may ask, how old are you? The photos on your web page look pretty young, like the one where you're playing the guitar. I'm 46.... For instance, I have learned that there is no known model of ZF. When Goedel & Cohen proved the independence of CH from ZF, they assumed that models existed. They didn't really construct models of ZF satisfying or violating CH. Goedel said assume any model of ZF, and he removed lots of sets in order for CH to be true. Cohen started with Goedel's model, and added lots of sets in order for CH to be false! I think this is fascinating, and I try to tell my pure Math buddies about this, and they're not interested, "Man we've got papers to write, we can't waste time on this nonsense!" ================================================================ 27 July 2004: E is a well-defined function (clearly total) by the ZFC subset or comprehension axiom: we can define sets (and therefore functions) by formulas that can involve quantifiers. Perhaps there's a better way of phrasing it. In the Math literature, this is standard, and no one would invoke any ZFC axioms to justify it.... ================================================================ 28 July 2004: Is it obvious that this map E is well-defined? If you've had pure Math courses (say real Analysis or point-set Topology), then you've seen this kind of function-construction, and much more complicated things I'm sure. But eventually, I say, you need the ZFC axioms to justify it. I'm having trouble with ZFC justification in general, myself, but it's a real priority of mine now to grok ZFC->Math. Anyway, my answer was: E is a well-defined total function by the ZFC subset or comprehension axiom: we can define sets (and therefore functions) by formulas that can involve quantifiers. ================================================================ 4 August 2004: ....It's all about Math....But it's not about constructing anything in a sense an engineer would understand.... ....R5RS DS makes heavier use of pure Math, by using Scott models, which are uncountable sets, and therefore couldn't be constructed by engineers. Heck, engineers can't even construct the set N of natural numbers, let alone its power set P(N), which is the Scott model, which also has a topology, i.e. an element of P(P(P(N))). With every P, we're an "order of magnitude" away from what engineers can build. But it's thought to be useful, in talking about programs. [Editor's note: The R5RS semantics has countable models. As for relevance of the power set, see the excerpt for 3 July.] ================================================================ 13 August 2004, addressing David Rush and speaking of a function that decides the halting problem: You may not know how to define such a function yourself, but take my word for it, it's much less hand-waving then Scott models. That it, it puts I'd say much less stress on the noncomputable ZFC axioms. [Editor's note: Those axioms might feel less stressed if Bill would give them a rest.] ================================================================ 26 August 2004, addressing Aatu Koskensilta: Back to models of ZFC: possible the speech I just gave would be false in an axiomatic ZFC proof. I'm way to ignorant to say. That's why I keep saying "pure Math is done in some model of ZFC." That way I don't have to know anything about ZFC! This might help: You and Lauri have both used the phrases "informal set theory" versus "formal set theory." I don't think I and my pure Math buddies are doing what you'd call informal set theory! Suppose we were invited to fancy dinner, and we showed up in tee-shirts, jeans & running shoes, maybe somebody'd ask us, where's your tux, and we'd say, we felt like dressing informally, our tuxes are kinda tight anyway. That is: we have the capacity to dress formally, but we didn't feel like doing it. But no suppose a homeless person shows us, dressed much worse than us. They're wearing their best clothes! They're not dressing informally! I'm thinking in particular of Vonnegut's character Kilgore Trout, maybe in Breakfast of Champions, who had to show up for a tux function, and I just remember the line, "Kilgore Trout hadn't owned a toothbrush in years." So, you & Lauri & Halmos maybe know enough ZFC to be able to formalize the pure Math world. That is, restate all the theorems, proof & constructions in axiomatic ZFC, so no models of ZFC ever come up. But I don't enough ZFC to formalize! My ignorance is general is the pure Math world. I'm working on it.... [Editor's note: Bill is saying that he and his pure Math buddies are bums, that informal set theory is the best they can do.] ================================================================