Counterintuitions and the well-ordering theorem
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From: Aatu Koskensilta on 27 Oct 2009 23:35 Herman Jurjus <hjmotz(a)hetnet.nl> writes: > Look, there may be any amount of valid reasons to be suspicious of AD, > or to dislike it, or whatever. But your objection is not one of them, > imho. A very simple reason to be suspicious of AD is that it amounts to a strong form of quantifier switch that is in general not valid. (I really couldn't make anything of your comments about the law of excluded middle, which isn't involved.) I'm utterly baffled by the idea that AD might be as evident as choice -- although of course it's a perfectly understandable position that both are horribly and equally dubious -- since for me choice is simply a mathematical expression of the idea that sets are arbitrary collections, an idea pleasing to the intellect, allowing me to make mathematical use of various intuitively palatable informal notions, conceptions, ideas, principles, while determinacy is an assertion about sets of reals about which I really have no intuitions whatever, certainly not merely on basis of my (mathematically very firm) grasp of the conceptual picture involved. (This is of course a very boring observation, since it amounts to merely stating I conceive of sets in such a manner as to make choice a triviality. The only interest in it in this context is that regarding AD we can't make the analogous boring observation. The observation holds for all the other usual axioms -- and various small large cardinal axioms -- as well, with the possible exception of replacement. Those who declare that choice, powerset, etc. aren't evident aren't presumably objecting to this, but to the conception involved itself, finding it "vague", "indeterminate", "theological", and so on. I base this on my experience that e.g. ardent and doctrinaire intuitionist are just as capable as I am of explaining why e.g. separation, infinity, and so on, naturally flow from the image of the world of sets provided by the narrative of the cumulative hierarchy. G�ran Sundholm once put it to me, that "There's too much slack in the classical 'meaning explanation' for set theoretic talk".) Your stance, that both choice and determinacy are true, calls for some further elucidation -- these principles are after all contradictory! The usual set theoretic "intuition" is that determinacy holds for some restricted class of sets, the obvious choice for this class being identified with L(R) very soon after the introduction of the axiom -- which Mycielski didn't propose as a truth about sets, but precisely as a principle that might be fruitfully studied, and hold, in context of some subuniverse of sets -- and this is precisely what is borne out in the study of large cardinals. Indeed, I think /both/ large large cardinals and (quasi-projective) determinacy derive their plausibility in a large extent from the various and systematic results about interconnections between these fields. (Kanamori and Moschovakis are excellent sources on this. And, Maddy's articles on set theoretic axioms haven't lost any of their currency by being less current owing to the passage of time.) -- 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 27 Oct 2009 23:42 Herman Jurjus <hjmotz(a)hetnet.nl> writes: > Well, don't look at me; i said right from the start that both AD and > AC are evidently true. What do you mean by this? On the face of it, it makes about as much sense as thunderously declaiming that there are infinitely many twin primes but there are only finitely many twin primes. Perhaps you hold some doctrine that set theoretic talk, in contrast to talk about naturals, say, is to be interpreted in some not-at-face-value manner, allowing for your stance, which, on a flat-footed, common-sense view, is rather odd? -- 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 27 Oct 2009 23:45 Butch Malahide <fred.galvin(a)gmail.com> writes: > On Oct 27, 11:17�am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: >> >> Well, don't look at me; i said right from the start that both AD and >> AC are evidently true. > > Not much use, then, in trying to use reductio ad absurdum on you, > then, is there? Actually, my comment was directed to Bill Taylor, who > is going to ignore it because he doesn't have a good answer. My apologies for butting in, but what was it that you asked? Bill is a sheep-shagging fence-sitter, and a notorious proponent of an apparently irremediably vaguely formulated form of "definitionalism" when it comes to sets. This with all respect to Bill, who will no doubt take it all in good humour. -- 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 00:58 Herman Jurjus <hjmotz(a)hetnet.nl> writes: > Aatu Koskensilta wrote: >> In contrast, choice seems to strike many as evident, both in the >> sense people explicitly state so > > Might i humbly suggest that that is explainable by the fact that many > people are confronted with choice in the earliest stages of their > studies, and not with AD? There's no question that many people explicitly state they find choice evident is because choice is a very commonly mentioned, invoked and treated a principle. I wonder, though, whether you meant to suggest people find choice evident on reflection and do not similarly find determinacy evident simply because they're exposed to choice earlier in their studies? This is a highly dubious claim. Set theorists have found, to various degrees, principles such as Vopenka's principle, the existence of a measurable, etc. compelling, on all sorts of grounds, but these people, who should know about this stuff, have never claimed to find determinacy similarly compelling, evident, clear, although many have of course offered the opinion that they think it's likely a correct principle for L(R). > And what would have happened if ZFC+AD had not been inconsistent? Or if > the inconsistency had not been discovered in the 1950s, but, say, in 2005? I'm not sure what's the relevance of this hypothetical. >> , and in the sense that arguments invoking (often >> implicitly) choice are regarded as convincing and compelling. (The >> latter observation is probably the more interesting one.) > > That's funny. I'd rather say that the interest in ZF-without-AC is > evidence for the contrary, namely the fact that practically all > arguments crucially requiring (full) AC give the impression of being > not only not convincing at all, but of being particularly > unenlightening, like the invoking of a magic stick. This is a pipe-dream. There's no widespread interest in ZF without choice, and most mathematicians are simply unable to even recognise invocations of choice -- as already mentioned, I doubt anyone who hasn't made it their business to study independence results in set theory realises the result that omega_1 has uncountable cofinality requires countable choice. Further, even though appeals to choice are occasionally noted, we do, by and large, accept as true a result if presented with a proof using (implicitly or explicitly) choice. (David's message on this, which accords with my experience, was very much to the point. Here we must of course stress, as I think you did, that usually implicit uses of choice reduce back, in so far as logical necessity is considered, to countable or dependent choice -- this is not surprising, since in ordinary mathematics we rarely meet stuff where these two principles wouldn't be sufficient. What we are make of this is an erudite matter, since a logical analysis, establishing whether this or that result logically requires choice, can't answer questions about whether the principle, and which form of the principle, was in fact "implicitly" used by this or that author; we need rather try and figure out what sort of modes of reasoning, ideas, plan of attack, guided the author in arriving at the proof -- I presented just this observation to Shapiro, when he gave a talk on "self-evidence" of various (set theoretic) axioms, and observed in particular it's obscure why we should see in this or that paper an "implicit" appeal to choice, and not dependent choice, or, say, global choice.) > It does often come in handy, so i do understand it's /popular/. But > 'convincing and compelling'? Yes, convincing and compelling. I have done some (very limited) empirical research on this, presenting choiceful proofs to logically innocent mathematicians and asking them if they find anything amiss. No-one objected to the constructions involving choice (even choice beyond dependent choice). I've had similar results with proofs using replacement. Usually, the lab rats were surprised when I explained the proofs involve (classically, and nowadays almost solely historically) controversial set theoretic principles. This goes with my long-held thesis, that the justification of the axioms of set theory does not come from the conceptual or philosophical analysis of the cumulative hierarchy -- this analysis or picture is simply an /explication/, not a justification -- but rather from the simple observation (found already in Peano) that these axioms simply say the set theoretic universe is closed under the operations mathematicians routinely (and without any worries or doubts) apply in their work (beginning with Peano, Cantor, Dedekind, etc.). >> 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? Very few, probably, if we understand by "people who accept AC" mathematicians who would not shy from invoking (often unwittingly) choice in their work. I'll now trot out an irrelevant hobby-horse. Set theorists are in the habit of talking about "equiconsistency results" etc. This is totally silly, since the results are /always/ stronger, establishing arithmetical conservativeness, strong reflection principles for set theoretic statements, and so on, on general grounds; we know, from our logical studies, even if it's not very often explicitly stated, that forcing, inner models, and all the methods, with the exception of G�delian stuff, we have at our disposal for proving independence results, equiconsistency results, and such like, always respect arithmetical truths, truth below a rank, and so forth and so on. There's not much hope of changing this ingrained terminology, but (again in my experience) it does actually lead to some confusion in the mind of the innocent student, and is something that, while appreciate by everyone who knows their set theory, is often missed by the less savvy outsider. > I would like to stress, though, that i didn't defend AD at the cost of > AC, but instead said that my intuitions regarding sets make AD and AC > /both/ come out as true. AC being acceptable a.o. on the grounds that > you mention above: it's arithmetically conservative. There is a very natural idea, that stuff in arithmetic, in finitary combinatorics, statements about objects in (intuitionistically acceptable, say) inductively defined classes in a general sense, are more determinate, objective, unambigous, than what we find in, say, higher set theory, by which we may here understand just the application of the "set-of" -operation a few times over the naturals, the reals, function spaces, groups, etc. It is surprisingly difficult to elucidate this natural attitude or idea, and explain in some detail just how talk of arbitrary sets of naturals is "more vague", "more indeterminate", "more theological" than talk of, say, arbitrarily large naturals. All these notions necessarily rely on our faculties of mathematical imagination, and unless we adopt some theses about these faculties, or their reach to some objective realm of mathematical objects, or their transcendental conditions, or whatever, it's difficult to see why, on any other than purely pragmatic grounds, the arithmetical conservativeness of choice should be an argument in its support. (Here by pragmatic grounds I have in mind just the purely mathematical observation that if e.g. a proof of Goldbach's conjecture should rely on choice, this reliance is necessarily inessential.) (Another complaint against set theorists: invariance under forcing, for the theory of hereditarily countable sets, for the next iteration, and so on, is often talked about in terms of "completeness" -- if we have enough large cardinals the theory of this or that mathematically important collection of objects is complete, in the sense that we can't disturb it by moving to a generic extension of the universe. This is a mathematically fruitful idea, and also very appealing, from an intuitive point of view, on which view mathematical statements we can't make true or false, willy-nilly, by model-theoretic means which don't really have any mathematically or conceptually obvious choice for "correctness", are somehow determinate, objective, what have you, in a contingent sort of way. But those of who are philosophically inclined, in the warped sense I am, naturally ask, Just why is invariance under forcing conceptually significant?) > Fwiw: the way i understand AD to be true is not incompatible with this > empirical observation. I'm afraid I don't understand your line of thought here. Does it not give you pause that AD implies the consistency (and all the arithmetical consequences, and more) of the large large cardinal axiom that there be infinitely many Woodin cardinals? Do you regard this fact as an argument for the existence (or arithmetical soundness) of an infinity of Woodin cardinals (perhaps, to borrow a phrase from Kanamori, "in the clarity of an inner model")? > Note, though, that the sample in this thread is not in accordance with > your empirical observation (four participants, two find AD compelling, > a third gives the strong impression that he simply hasn't given it a > good thought, yet.) My opinions and intuitions on these matters are of very little interest, and so are, sadly, those of the others who have said their say here -- we are not set theorists, and not a representative sample of mathematicians, and so a brute fact is that our opinions carry very little weight. I may be presumptuous in assuming that I've probably gone through more large cardinal arcana (and descriptive set theory arcana) than the other participants, but, alas, even this doesn't really entitle me to have any particularly informed opinion or "intuition" on the evidence of AD, Woodin cardinals, what not. Thus my coy insistence on deferring to what people in general, and what set theorists in particular, have to say about AD and AC, and what is implicit in their thinking and mathematical work. >> I see I have a sizable backlog of messages to address on sci.logic, so > > Apparently you get more sci.logic traffic than we do. What's your > newsserver? I have a mobile Internet connection through DNA, and apparently it provides a newsserver courtesy of Saunalahti (or Elisa). I really must answer a few of Daryl's posts, having detected a hint of exasperation in some of his replies, and having committed to the eternal archives of Usenet a few false claims, which claims Daryl found objectionable, owing to which state of affairs I should publicly flagellate myself. I seem to recall there were a few posts by Nam, too, that require my urgent attention -- in particular a baffling argument, according to which there's something wrong (and "non-syntactical") in the perfectly fine finitist proof of the consistency of Robinson arithmetic in Shoenfield. As you can surely appreciate, these matters Usenetical can't possibly be glossed over. > Yah; it doesn't seem to make sense to continue much further with this. Indeed. I trust my continuing much further with this pleases or vexes you no end. -- 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 01:17
On Oct 27, 10:45 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Butch Malahide <fred.gal...(a)gmail.com> writes: > > On Oct 27, 11:17 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > > >> Well, don't look at me; i said right from the start that both AD and > >> AC are evidently true. > > > Not much use, then, in trying to use reductio ad absurdum on you, > > then, is there? Actually, my comment was directed to Bill Taylor, who > > is going to ignore it because he doesn't have a good answer. > > My apologies for butting in, but what was it that you asked? Bill is a > sheep-shagging fence-sitter, and a notorious proponent of an apparently > irremediably vaguely formulated form of "definitionalism" when it comes > to sets. This with all respect to Bill, who will no doubt take it all in > good humour. 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)? |