Counterintuitions and the well-ordering theorem
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From: Bill Taylor on 29 Oct 2009 23:30 OHerman Jurjus <hjm...(a)hetnet.nl> wrote: > But: either player 1 has a winning strategy or he hasn't. > Now what does it mean for player 1 to not have a winning strategy? > > I'd say that amounts to 'player 2 has some way to prevent player 1 from > winning'. That sound fairly unimpeachable. And as *someone's* got to win every time, (LEM?), preventing the opponent is winning. > It's a bit like with the Jordan curve theorem: it's nice that we can > prove it, but had our definitions been such that it had come out as > false, we would only have concluded that our definitions needed > revision, not that the Jordan curve theorem is false. A very apposite comment indeed! -- Beaming Bill
From: Bill Taylor on 29 Oct 2009 23:33 > It's a bit like with the Jordan curve theorem: it's nice that we can > prove it, but had our definitions been such that it had come out as > false, we would only have concluded that our definitions needed > revision, not that the Jordan curve theorem is false. And I meant to add - this was what was intended to happen with the Banach-Tarski decomposition, or so I have read. Allegedly, one of them hoped it would be so ludicrous that mathies as a whole would thus drop AC, but that didn't happen. They'd forgotten that mathies are very like Alice's Red Queen - "Why, I generally believe six impossible things every day before breakfast!" -- Beliefless Bill
From: Bill Taylor on 29 Oct 2009 23:42 Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Why shouldn't we spend extraordinary amounts of energy on things we > don't take too seriously? Surely it's a dismal, truly teeth-gnashing > inducing idea, a revolting notion fit only to be rejected in a violent > explosion of metaphorical vomit, that we only spend considerable length > of time on stuff that we take very very seriously. I agree totally with this! As Bertie Russ said, "The time you enjoy wasting is not wasted time!" > I also find that I actually learn something of these electronic exchanges, > coming out of them illuminated with new wisdom ... ... not > necessarily anything of mathematical nature, but something of how > people, myself included, react to this or that mode of argumentation, > this or that way of putting this or that, this or that line of thought, > this or that level of formality, gathering in the process valuable > information about whether this or that way of putting an idea is > generally intelligible, gaining for myself many a curious factoids about > this or that English idiom and its use and abuse, what have you. Or this > or that. Bits and pieces, follies and human insight. That's what I reap > from these virtual encounters. Exactly so! Very well put! Agree 100%. A keeper. > It is also my hope that, in spite of my sometimes needlessly aggressive > debating style, and peculiar and failed attempts at humour, those with > whom I battle wits leave these Usenetical battlefields slightly > improved, with a perspective on life just a whit expanded, ... > from what it was before. I'm sure we all earnestly hope that. > It is customary in many newsgroups to include in otherwise off-topic > posts a nugget of topicality. Here goes: And here's mine: In the absence of AC, CH can be seen to be both true and false, depending on how it is worded: 1) There is a subset of R, of cardinality strictly between N and R. 2) There is a function on subsets of R whose values are bijections between the argument and one of N_k, N, or R. AC |- 1 == 2 But, definitionially speaking, 1 and 2 are both clearly false. And they might both be false in models of ZF. -- Troublesome Taylor
From: Bill Taylor on 29 Oct 2009 23:49 Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Rupert <rupertmccal...(a)yahoo.com> writes: > > When I first read about AD I thought it was an interesting hypothesis, > > but I never had any feeling that it was intuitively plausible. Different strokes, different folks. On my first hearing of it, it struck me as *obviously* true, and a game-theoretic necessity. > > On the other hand as soon as I encountered AC > >I was completely convinced that it was true. As was I. Like believing in God. It was only much, much later that I realized how I'd been cleverly hoodwinked! > Our agreement on these matters is most touching. Let's hug! Please, there are children on this newsgroup!! > I soon learned determinacy was intended to apply to sets that behave, > sets that are in some intelligible manner built of basic, familiar and > cozy base by means of (transfinitely iterated) operations that make > (comfortable, familiar, cozy, mathematical) sense. That must have been very cheering for you, like finding that heaven was only intended for right-thinking folks, and not very bad boys! To me, though, the nature of the sets it applies to is wholly irrelevant. The Lord welcomes even the most grievous sinners! -- Welcoming William
From: Bill Taylor on 29 Oct 2009 23:52
stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >It's a bit like with the Jordan curve theorem: it's nice that we can > >prove it, but had our definitions been such that it had come out as > >false, we would only have concluded that our definitions needed > >revision, not that the Jordan curve theorem is false. > > Yes, I agree completely in this case. I feel that if it is false, > then it means we have defined "continuous curve" incorrectly. We all seeem to agree here. > As a matter of fact, I think it is provably *false* for some natural > ways of formulating continuity. Whoa! That IS interesting! Would you do us all a kindness, Daryl, and teach us about some of these natural alternatives. Perspiring minds want to know! -- Breathless Bill |