Counterintuitions and the well-ordering theorem
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From: Aatu Koskensilta on 28 Oct 2009 01:52 Butch Malahide <fred.galvin(a)gmail.com> writes: > Observing that, unlike choice, which can hold "all the way up", > determinacy can only hold for games of limited size, I wanted Bill (or > someone who claims that AD is "intuitive") to explain, if it is > "intuitively obvious" that determinacy holds for games where a play is > a sequence of natural numbers (or real numbers), why the same > intuition does not lead (falsely) to determinacy of games where a play > is a sequence of countable ordinals (or sets of real numbers)? Well, I couldn't really make any of the arguments for determinacy that were presented. But your observation is related to the only instance of anyone in the literature proposing an (informal) argument for the truth of determinacy. The idea, as far as I've managed to make out, is that we have two "infinitely clever" players, who know all facts about (sets of) reals, and so should be able to use this their infinite wisdom to win any winnable game. This idea (which, I think, was, probably whimsically, put forth by Mycielski and Steinhaus) is obviously wrong-headed since it obviously implies /all/ games, whatever their nature, and lofty location in the set theoretic hierarchy, are determined. In all fairness to the AD proponents in this thread, it should be observed that Rheinhardt's arguments for the "ultimate large cardinal axiom", that there be a non-trivial elementary embedding of the set theoretic universe in itself, were pure waffle of similarly exacting exactitude. (Maddy puts this somewhat more diplomatically in her /Believing the Axioms/ papers...) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Butch Malahide on 28 Oct 2009 02:20 On Oct 28, 12:52 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Butch Malahide <fred.gal...(a)gmail.com> writes: > > Observing that, unlike choice, which can hold "all the way up", > > determinacy can only hold for games of limited size, I wanted Bill (or > > someone who claims that AD is "intuitive") to explain, if it is > > "intuitively obvious" that determinacy holds for games where a play is > > a sequence of natural numbers (or real numbers), why the same > > intuition does not lead (falsely) to determinacy of games where a play > > is a sequence of countable ordinals (or sets of real numbers)? > > Well, I couldn't really make any of the arguments for determinacy that > were presented. But your observation is related to the only instance of > anyone in the literature proposing an (informal) argument for the truth > of determinacy. The idea, as far as I've managed to make out, is that we > have two "infinitely clever" players, who know all facts about (sets of) > reals, and so should be able to use this their infinite wisdom to win > any winnable game. This idea (which, I think, was, probably whimsically, > put forth by Mycielski and Steinhaus) is obviously wrong-headed since it > obviously implies /all/ games, whatever their nature, and lofty location > in the set theoretic hierarchy, are determined. In all fairness to the > AD proponents in this thread, it should be observed that Rheinhardt's > arguments for the "ultimate large cardinal axiom", that there be a > non-trivial elementary embedding of the set theoretic universe in > itself, were pure waffle of similarly exacting exactitude. (Maddy puts > this somewhat more diplomatically in her /Believing the Axioms/ > papers...) You mean William Reinhardt, not "Rheinhardt". Not a spelling flame, just trying to be helpful in case anyone reading this hasn't heard of Reinhardt before and wants to look up his work. Thanks for your reply.
From: Aatu Koskensilta on 28 Oct 2009 02:28 Herman Jurjus <hjmotz(a)hetnet.nl> writes: > Aatu Koskensilta wrote: > >> Our ideas about what is or is not evident are of course not >> arbitrary, and we can bring to bear considerations of less subjective >> or relative character. For example, we may note that choice is an >> innocent principle in a quasi-Hilbertian sense, in that it doesn't >> have any "concrete" consequences, while determinacy is a (moral) >> large cardinal axiom, of staggering consistency strength, and in >> particular implies all the arithmetical consequences of "there are >> infinitely many Woodin cardinals". > > How many people who accept AC (and reject AD) know the above, and for > how many is/was it an argument? Should it be? I forgot to expound in detail (ha-ha!) on your last question. What I wrote above is a mouthful, and I'm unsure just what you're asking should or should not be an argument, and for what. I already indirectly expressed the opinion that the arithmetical conservativeness -- the conservativity results can be extended a few levels up in the analytic hierarchy, using Shoenfield's absoluteness lemma and a few more refined devices -- of choice isn't a good argument for choice. This is mainly because mathematicians virtually never appeal to this fact in their proofs (and, as you so shrewdly imply, are very rarely even aware of the fact) or in their accounts of their mathematical dealings with sets. In light of this it would be odd to regard the conservativity as an argument for choice, at least in any sense of "argument" that has anything to do with actually convincing people.[1] AD's strength is, on the other hand, something I'd expect anyone who's even superficially acquainted with these matters to be fully aware of. And it's simply impossible to find any mathematician unwittingly relying on AD to prove this or that. My comment above was, to a large extent, subjunctive [my sense of English says this is a very odd choice of word, but it seems to be in accord with at least pretentious academic usage] in that I would simply surmise that if someone were, by their keen intuition, by "a sort of persuasion", by being hit on the head with a brick, immediately struck with the obvious truth of determinacy, the observation that the arithmetical soundness of an infinity of large large cardinals follows would probably give them pause. (After all, most people who regard choice as evident, acceptable, obvious, something it's interesting to study the consequences of, are comforted by the (commonly half-digested) knowledge it's relatively consistent. And most people are, often for no deep reason, but simply for not having been exposed to such matters, suspicious of large cardinals, let alone large large cardinals. This comment I again base on the observation that people often go on about such matters in a most naive and innocent manner. as you yourself have no doubt witnesses in news, and in other venues.) Footnotes: [1] There was, several years ago, an amusing thread on the FOM mailing list, in which Stephen Simpson would insist that something may well be of "general intellectual interest" even if, in fact, it is of absolutely no interest to anyone. Needless to say, there's no need to take this sort of "objectivism" very seriously. Arguments are directed at people, evidence is something we experience or fail to experience, stuff is interesting to the extent we actually find it interesting... This doesn't of course preclude the possibility that something that at first strikes us as boring is later found to be very interesting, its potentially opaque connection to what we take interest in having been made more apparent; that we come to recognise something as evident after it's explained to us it's just another way of saying something we already unhesitantly accept, a trivial corollary thereof, etc. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 28 Oct 2009 03:22 Butch Malahide <fred.galvin(a)gmail.com> writes: > You mean William Reinhardt, not "Rheinhardt". Not a spelling flame, > just trying to be helpful in case anyone reading this hasn't heard of > Reinhardt before and wants to look up his work. Right you are, of course, it's "Reinhardt"! My mistake is baffling and inexcusable -- composing the reply I did in fact consult Part II of _Axiomatic Set Theory_ /Proceedings of Symposia in Pure Mathematics/, vol. XIII Reinhardt article. In the two volumes (the publication separated by a few years, if I'm not mistaken), we find a true cornucopia of wonderful papers, including, taking out a few random titles from Part I: /Sets constructible using L-kappa-kappa/ by Chang [A very short paper, consisting almost entirely of definitions. But we all know sometimes the right definitions are the crux of the matter! Armed with the right definitions, the right concepts, we practically immediately, for free, obtain strong generalisations of familiar results.] /Comments on the foundations of set theory/ [A very silly paper, demonstrating very well the well-known truism that even mathematicians who have done seminal work in foundations can blather boringly, piling a platitude on a platitude, and on the rare moment of making a substantial assertion -- being guilty of the sin of utmost, idiotic philosophical naivete. But, then, this might well be said of some passages in G�del...] /Sets, semisets, models/ by Petr Hajek [This paper is almost as unreadable as the book. Great fun for those who have a penchant for formalism.] /Primitive recursive set functions/ by Jensen and Karp [Most later stuff basically goes back to this paper, in so far as essential definitions are concerned, and which includes almost all the modern simplifications.] /Observations on popular discussions on foundations/ by Kreisel [Kreisel never disappoints those of us who like perceptive and well-deserved potshots at philosophically naive people -- in this instance at Cohen and Robinson; here we also find the famous observation that second-order set theory decides the continuum hypothesis, plus a few memorable Kreiselisms...] /On the logical complexity of several axioms of set theory/ by Levy [No need to say anything about this, ] /On some consequences of the axiom of determinateness/ by Mycielski [When did the axiom become that of "determinacy"? In many places where Kechris's paper on dependent choice holding in L(R) assuming AD is mentioned we find determinacy spelled "determinancy". What's with this orthographic chaos?] /Ordinal definability/ by Myhill and Scott [What's the G�del piece where this notion was first introduced, only to be completely ignored for years?] /Unramified forcing/ by Shoenfield [Still one of the best expositions of forcing, but possibly not for the faint of heart who would like, say, some motivation for the notion of generic set.] /Consistency of GCH with the existence of a measurable cardinal/ [Off with Cohen's reliance on constructibility!] /Real-valued measurable cardinals/ by Solovay [Everyone who's heard about measurable cardinals should naturally wonder about /real-valued/ measures... Much more stuff in the paper than the title indicates.] /Transfinite sequences of axiom systems for set theory/ by Sward [By all accounts a rather inconsequential paper. Included here only because it has a (somewhat strenuous) connection with my own research (carried out years ago, in a mental institution)!] Of the times (that is, the times a few years before the symposion) James E. Baumgartner writes in his review of Kunen: Once upon a time, not so very long ago, logicians hardly ever wrote anything down. Wonderful results were being obtained almost weekly, and no one wanted to miss out on the next theorem by spending the time to write up the last one. Fortunately there was a Center where these results were collected and organized, but even for the graduate students at the Center life was hard. They had no textbooks for elementary courses, and for advanced courses they were forced to rely on handwritten proof outlines, which were usually illegible and incomplete; handwritten seminar notes, which were usually wrong; and Ph.D. dissertations, which were usually out of date. Nevertheless, they prospered. Now the Center I have in mind was Berkeley and the time was the early and middle 1960's, but similar situations have surely occurred many times before. In this case, to the good fortune both of the graduate students and of the logicians not lucky enough to be in California, all the wonderful results were eventually written down. But it took a surprisingly long time. Ah, to be young again, and at a different age, at a different place. Though, in all likelihood, I would have been just extremely depressed by everyone else around being so extremely gifted and clever. (A reaction Dana Scott, who in this context should be credited as a co-creator of the (elegant but practically useless) Boolean-valued formulation of forcing, tells us he had at the time. This piece of trivia might, in some small way, be of some consolations to us mere mortals, who probably don't ever get to prove things as exciting as that the reals may be a countable union of countable sets, or that collapsing felicitously chosen cardinals introduces all sorts of interesting structure in the continuum and sets thereof...) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Herman Jurjus on 28 Oct 2009 03:31
Aatu Koskensilta wrote: > if someone were, by their keen intuition, by "a sort > of persuasion", by being hit on the head with a brick, immediately > struck with the obvious truth of determinacy, the observation that the > arithmetical soundness of an infinity of large large cardinals follows > would probably give them pause. If we accept both AD and AC as true, then we obviously have to give up some of the other assumptions, either some of the other axioms of ZF, or some of the inference rules of FOL, or some silent assumption that we're not even aware of yet. Once we've identified and eliminated it/them, who says that all consequences of ZF+AD will remain provable? -- Cheers, Herman Jurjus |