Counterintuitions and the well-ordering theorem
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From: Herman Jurjus on 26 Oct 2009 08:10 Daryl McCullough wrote: > Bill Taylor says... >> The first description makes it "intuitively clear" that >> the game should always be "determined", that is, have >> a winning strategy for one or the other player, as at >> every step it is a game of complete information. > > But *why* does a game of complete information imply > that there is a winning strategy for one player or > the other? That's a *theorem* that must be proved. Perhaps the point is not so much that it is completely self-evident that AD is true, but that our reasons for believing AC (or power set) are at least as superficial and shaky, if not more so. (Personally, i can only make sense of sets larger than P(N) when i go into 'sloppy reasoning mode'. And in this sloppy reasoning mode, AD is at least as plausible as AC.) -- Cheers, Herman Jurjus
From: Aatu Koskensilta on 26 Oct 2009 14:07 Herman Jurjus <hjmotz(a)hetnet.nl> writes: > Perhaps the point is not so much that it is completely self-evident > that AD is true, but that our reasons for believing AC (or power > set) are at least as superficial and shaky, if not more so. What is evident to some might be completely opaque to others. Often all we can say about such things is that many people do in fact find this or that evident. This is certainly the case with the axiom of choice, but not with determinacy. I can think of only one instance in the literature of anyone arguing that determinacy can be seen to be true on basis of informal reflection (this reflection having to do with "infinitely intelligent" players and what not). In contrast, choice seems to strike many as evident, both in the sense people explicitly state so, 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.) 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". Such considerations are by no means conclusive -- if someone is sufficiently impressed by the evidence of determinacy they will naturally regard this as powerful evidence for (at least the arithmetical consequences of) infinitely many Woodin cardinals! Again, it is an empirical observation that no set theorist seems to take this view. I see I have a sizable backlog of messages to address on sci.logic, so I'll limit general blather about evidence to these haphazard remarks. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Herman Jurjus on 27 Oct 2009 10:03 Aatu Koskensilta wrote: > Herman Jurjus <hjmotz(a)hetnet.nl> writes: > >> Perhaps the point is not so much that it is completely self-evident >> that AD is true, but that our reasons for believing AC (or power >> set) are at least as superficial and shaky, if not more so. > > What is evident to some might be completely opaque to others. Often > all we can say about such things is that many people do in fact find > this or that evident. This is certainly the case with the axiom of > choice, but not with determinacy. I can think of only one instance in > the literature of anyone arguing that determinacy can be seen to be > true on basis of informal reflection (this reflection having to do > with "infinitely intelligent" players and what not). 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? 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? > , 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. It does often come in handy, so i do understand it's /popular/. But 'convincing and compelling'? > 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 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. > Such considerations are by no means conclusive -- if > someone is sufficiently impressed by the evidence of determinacy they > will naturally regard this as powerful evidence for (at least the > arithmetical consequences of) infinitely many Woodin cardinals! Again, > it is an empirical observation that no set theorist seems to take this > view. Fwiw: the way i understand AD to be true is not incompatible with this empirical observation. 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.) > 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'll limit general blather about evidence to these haphazard remarks. Yah; it doesn't seem to make sense to continue much further with this. -- Cheers, Herman Jurjus
From: Herman Jurjus on 27 Oct 2009 10:04 Daryl McCullough wrote: > Bill Taylor says... > >>> I know that you're not offering that as a mathematical argument, >>> but I would like to understand your intuitions here. >>> "Either player 1 has a winning strategy, or 2 has a defense" >>> needs a little more argument to be compelling, because of games >>> like "Rock, paper, scissors" with no strategy guaranteed to win. >> OUCH! Daryl, this is not up to your usual high debating standards. >> RockPS, like almost any game of incomplete information, has no >> "guaranteed winning strategy", as you say, but AD games are >> not of this type. > > Actually, could you explain in what sense chess or checkers are > games of complete information, and rock/paper/scissors is *not*? Now it's my turn to say 'OUCH!' Can you explain in what way chess or checkers are /not/ games of complete information? > In all cases, each player must make his move in ignorance of > what move his opponent will make. There is no "hidden state" > (such as the hidden playing cards in a game of cards), unless > you count the intentions of your opponent as hidden state, in > which case, all games are games of incomplete information. Only if you mis-characterize them as 'both players simultaneously choose a strategy, and keep following these'. But that's not what happens when two people play chess. It really /does/ make a big difference whether a game is of complete information or not. And it's not a coincidence that the (finite) games of complete information are all determined. Likewise, it's not a coincidence that it's non-trivial to find concrete (i.e. ZF definable) counterexamples against AD. 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. -- Cheers, Herman Jurjus
From: Herman Jurjus on 27 Oct 2009 10:04
Butch Malahide wrote: > Can you explain why determinacy is intuitively clear when each move is > choosing a natural number, but no longer clear when each move is > choosing a countable ordinal? or a set of real numbers? > > What does your intuition say about the following game? First, White > chooses a set X or real numbers; then Black chooses a real number x. > Black wins if either X is empty or x is in X; White wins if X is > nonempty and x is not in X. I'm sure you will agree, Bill, that > neither player has a winning strategy. What gives? Huh? Black clearly has a winning strategy, no? -- Cheers, Herman Jurjus |