Counterintuitions and the well-ordering theorem
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From: Daryl McCullough on 19 Nov 2009 13:03 George Greene says... > >On Nov 18, 12:24=A0pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> The second player doesn't *KNOW* what strategy the first player >> chose. All that he knows is the first *MOVE* in that strategy. > >Well, your use of the word "defense" was confusing. Sorry. Here's the whole gory detail: 1. A "game" (of the type we are discussing) is defined by a set G of infinite sequences of naturals (we identify an infinite sequence of naturals with a function from omega to omega). 2. A "strategy" is a function f that takes a finite sequence of naturals and returns a natural (we identify a finite sequence with a function from a finite initial segment of omega to omega). 3. Given two strategies f and g, define play(f,g) to be the sequence of naturals s such that s(0) = f([]) s(2n+1) = g([s(0), ..., s(2n)]) s(2n+2) = f([s(0), ..., s(2n+1)]) where [a_0, a_1, ..., a_n] means the sequence s such that s(j) = a_j. 4. Define beats(f,g,G) as: play(f,g) is an element of G 5. f is a winning strategy for the first player if: forall g, beats(f,g,G) 6. g is a winning strategy for the second player if: forall f, ~beats(f,g,G) 7. There is no winning strategy for the first player if: forall f, exists g, beats(f,g,G) 8. There is a winning strategy for the second player if: exists g, forall f, beats(f,g,G) Because of the order of the quantifiers, the lack of a winning strategy for the first player does not logically imply there is a winning strategy for the second player. -- Daryl McCullough Ithaca, NY
From: Charlie-Boo on 19 Nov 2009 14:38 On Oct 12, 1:16 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > It is often said that the well-ordering theorem, or the existence of a > well-ordering of the reals in particular, is counterintuitive. Alas, > I've never quite fathomed what intuitions are contradicted. Perhaps > someone with keener intuition into the mysteries of sets can shed some > light on this pressing matter? LT(I,x) ^ (all A)~LT(I,A)v~LT(A,x) is not defined on the reals where I = input, x = output, and LT(x,y) means x<y, mostly because there's no such A. C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon mann nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Jesse F. Hughes on 19 Nov 2009 15:33 Charlie-Boo <shymathguy(a)gmail.com> writes: > On Oct 12, 1:16 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> It is often said that the well-ordering theorem, or the existence of a >> well-ordering of the reals in particular, is counterintuitive. Alas, >> I've never quite fathomed what intuitions are contradicted. Perhaps >> someone with keener intuition into the mysteries of sets can shed some >> light on this pressing matter? > > LT(I,x) ^ (all A)~LT(I,A)v~LT(A,x) is not defined on the reals where I > = input, x = output, and LT(x,y) means x<y, mostly because there's no > such A. Er, right. Good point. Keen grasp of the well-ordering theorem, there. You *do* realize that the well-ordering theorem entails that there is a well-ordering of R, not that the standard order on R is a well-order, right? That is, there is a relation S c R x R such that S is a well-ordering. Clearly, the standard order < on R is not a well-ordering. -- Jesse F. Hughes "To be honest, I don't have enough interest in math to spend the time it would take to clean up the mess that I believe has been created in the past 100 or so years." -- Curt Welch lets the world down.
From: Daryl McCullough on 19 Nov 2009 16:31 In article <fc751abe-12a5-45e6-b2b0-0080e2de9104(a)a21g2000yqc.googlegroups.com>, George Greene says... > >On Nov 12, 1:23=A0pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: > >> People skeptical of the power set usually balk at the very first >> application that gets you something really new: P(omega), or the >> reals. > >The other time to balk is at the very end, because there in fact >is no end. This is a place&case where people could legitimately >talk about potential infinity. There is no last ordinal and there is >no ultimate universe of all powerclasses. I think that's an interesting point. It's actually a kind of odd situation. There is a sense in which sets cannot be thought of as a "completed totality", which is the reason that there cannot be a set of all sets. On the other hand, we can by fiat declare some new object (not in ZF, but in Morse Kelly set theory or in Quine's NF set theory) to be the complete collection of all sets, and we can consistently deal with such a completed collection. So what exactly, is the sense in which the collection of sets cannot be completed? -- Daryl McCullough Ithaca, NY
From: Bill Taylor on 20 Nov 2009 00:32
stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > I guess I'm confused about the ontological status of abstract >objects such as sets.It seems silly to think of them realistically, > as if there exists, in some Platonic heaven, the universe of sets, > in which all questions, such as the existence of msrbl cardinals, > is resolved. On the other hand,it also doesn't seem right to treat > them as pure fictions, where the question of "Does there exist > a cardinality between N and P(N)?" has no more of an answer than > "What's the name of Sherlock Holmes' grandmother's dog?" Well, I'm surprised to hear you admit as much, considering the thread so far! Yes, the ontological status [O.S.] of (some) sets is very murky, though we may have no qualms about the ontological status of simpler abstract objects such as 1,2,3... a,b,c,... and even N and its various equivalents. What you have said thus provides a motivation for the earlier question whose response you claimed a cop-out. You admit doubts about the O.S. of sets; I presume (maybe wrongly?) you have no, or at least much lesser, doubts about the O.S. of natural numbers. If this is so, then my own suspicions about powerset etc, are motivated by similar concerns. We have no doubt about individual natural numbers, or even N, because we feel that we know "all about them" in some sense. Though various theorems still surprise us, we feel that there can be no further *philosophical surprises* from them, if that phrase strikes a chord. I have long been exercised by the concern, how far can we extend this to SETS of naturals, i.e. to real numbers? What reals numbers (and beyond) are in some sense INEVITABLE in math, regardless of the framework we choose to "run" math on top of. We both admit (do we?) the inevitability of naturals, and N, and its equivalents. We also must admit (IMHO) the inevitability of *certain* reals, such as sqrt(2), pi, e, and vastly many more; all the individual recursive reals, at least. [ To attach to another concern, I might say FOR ANY recursive real, that real is inevitable, but that FOR ALL recursive reals, they may not be, (i.e. the set of such may not be). ] And if it turns out that it is OK, there are still the r.e. reals, the definable reals, and so forth, all with graver suspicions about their "reality", their OS, that is their inevitability independent of the way math is framed in substrate. stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > I guess I'm confused about the ontological status of abstract >objects such as sets.It seems silly to think of them realistically, > as if there exists, in some Platonic heaven, the universe of sets, > in which all questions, such as the existence of msrbl cardinals, > is resolved. On the other hand,it also doesn't seem right to treat > them as pure fictions, where the question of "Does there exist > a cardinality between N and P(N)?" has no more of an answer than > "What's the name of Sherlock Holmes' grandmother's dog?" Well, I'm surprised to hear you admit as much, considering the thread so far! Yes, the ontological status [O.S.] of (some) sets is very murky, though we may have no qualms about the ontological status of simpler abstract objects such as 1,2,3... a,b,c,... and even N and its various equivalents. What you have said thus provides a motivation for the earlier question whose response you claimed a cop-out. You admit doubts about the O.S. of sets; I presume (maybe wrongly?) you have no, or at least much lesser, doubts about the O.S. of natural numbers. If this is so, then my own suspicions about powerset etc, are motivated by similar concerns. We have no doubt about individual natural numbers, or even N, because we feel that we know "all about them" in some sense. Though various theorems still surprise us of course, we feel that there can be no further *philosophical surprises* from them, if that phrase strikes a chord. I have long been exercised by the concern:- how far can we extend this idea to SETS of naturals, i.e. to real numbers? What reals numbers (and beyond) are in some sense INEVITABLE in math, regardless of the framework we choose to "run" math on top of. We both admit (do we?) the inevitability of naturals, and N, and its equivalents. We also must admit (IMHO) the inevitability of *certain* reals, such as sqrt(2), pi, e, and vastly many more; all the individual recursive reals, at least. [ To attach to another concern, I might say FOR ANY recursive real, that real is inevitable, but that FOR ALL recursive reals, they may not be, (i.e. the set of such may not be). ] And if it turns out that it is OK, there are still the r.e. reals, the definable reals, and so forth, all with graver suspicions about their "reality", their OS, that is their inevitability independent of the way math is framed in substrate. If you are still following me, that is the motivation for my suspicions as expressed earlier, where I said that if you have no feeling for it already, then no debate could give you one. From your topmost remarks, it seems as if you MAY have a glimmer of such feelings? > ..."every set has a power set", but beyond > consistency, I don't really understand what it *means* to doubt > its truth. It's not so much doubt about truth, as a choice of WHAT TYPE of set theory we are going to entertain, and work in. Let me propose a hopefully apposite parallel. For a while there was a lot of stir about Aczel's non-well-founded sets. People were momentarily exercised by the though of "whether they existed" or not. But it soon died away - consistency is sufficient to make them worth studying, and as to whether they exist otherwise - who cares? But the math world soon relegated them to little more than a curiosity, saying (in effect) "they may exist BUT we're going to henceforth work in that subcollection of them that ARE well-founded". And that's all that happened. Now the same *could have* occurred with these excessively Cantorian sets that I'm suspicious of. Poincare famously hoped it would be so! And no great harm would have come to math if it HAD been so - all our usual branches of everyday math would have been much the same, right through C20 math, though admittedly rather more cumbersomely and scrappily presented, with little overlying unity or elegance. And that's the key! Mathies, especially pure mathies, are totally enraptured by unity and elegance - and that's why ZF(C) now reigns supreme. NF and Aczel have been ignored, and any "definitionalist" approach has been still-born through lack of proper pre-natal care. I suggest it may be time to see if the baby can be revived. > Does it really make sense to doubt the truth of the > claim that there is a number i such that i*i = -1? That's totally different and completely unobjectionable. Complex numbers can be defined trivially as ordered pairs with the appropriate operations, and all the usual results obtained. It has no logical or philosophical problematicity at all - i.e none beyond whatever the reals that make them up, already have. > We essentially *define* i into existence by that claim. If you like. But it's still intriguing to note, that you *haven't* really done so - because there's NO WAY you can distinguish between i and (-i) ! You could interchange them in the whole of math, and no-one would ever know! But all this is peripheral. > It seems to me that the > power set of omega is similarly defined into existence. It is a totally different question in every respect. -- Bulldozing Bill ** The math is done right, but is the right math done? |