Counterintuitions and the well-ordering theorem
in [Logic]
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From: George Greene on 18 Nov 2009 10:56 On Oct 23, 10:39 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > "Either player 1 has a > winning strategy, or player 2 has a defense" needs a little more argument > to be compelling, because of games like "Rock, paper, scissors" where > no strategy is guaranteed to win. > What's the argument against the > possibility that (1) for every strategy for the first player, there > is a defense for the second player, and (2) for every defense for the > second player, there is a strategy for the first player that beats it. This just irrelevant if true. The 2nd player will always choose the successfully defending strategy. The fact that the 1st player COULD have chosen a strategy that would beat this one IS IRrelevant, because IF the 1st player had chosen THAT strategy, the 2nd would've chosen the one that defended against THAT strategy (rather than the one that defends against this one). > I don't see any reason to believe that this CAN'T be the case. Well, there is, but it has more to do with perfect information and the fact that "1st" and "2nd" are just not constant -- AFTER the 1st player has gone, the 2nd player IS 1st, for NOW.
From: George Greene on 18 Nov 2009 11:00 On Nov 12, 1:23 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > But what does that really mean? What does it mean to say that > the collection of all reals (which exists in the sense that it > is a definable class) does not exist *as* *a* *set*? It seems > to me that we only accept the set/class distinction because > it is *forced* on us by consistency. This is an amazingly odd locution. Things that are "forced on us by consistency". i.e., things that would lead to inconsistency if we didn't accept them, are NORMALLY considered PROVED!! That is a DEFINITION of what it MEANS to prove something!! If something is "forced on us by consistency" THEN IT *IS REAL*. It is even TRUER THAN"true" because it is NECESSARILY true. It is true in ALL POSSIBLE models of whatever area you are trying to investigate.
From: George Greene on 18 Nov 2009 11:02 On Nov 12, 6:01 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > Unrestricted comprehension gets you into trouble, right? Comprehension IS unrestricted TO BEGIN with in the sense that even if you do NOT have a SET for every formula, YOU STILL have a CLASS for every formula (it's just that the class may be proper). If you go up to a class theory then the "restriction" involves what you can quantify over, so it is more unrestricted quantification than unrestricted comprehension that is implicated -- you are not supposed to quantify over a universe including the thing BEING defined.
From: George Greene on 18 Nov 2009 11:05 On Nov 17, 6:53 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > But all it takes to be a class is a criterion for membership. A class > is basically a formula with a single free variable. There is no question > that the *formula* exists---we can write it down. There IS SO TOO a question about what exists, at FIRST order. In THAT semantic paradigm, "exist" is a totally technical and counter- intuitive term -- what exists is what's IN THE DOMAIN getting quantified over -- EVERYthing else, VERY much INCLUDING ALL "formulas" (unless you are theorizing About formulas) Does NOT Exist! > It exists as a formula, as a criterion for separating reals from > non-reals. What else do you expect from a "collection"? I > really don't know what you mean by a collection if not that. Nobody is trying to define "collection" at FIRST order! At FIRST order we quantify over INDIVIDUALS! If you want to quantify over COLLECTIONS then you basically MUST go to SECOND order! This largely defeats the whole computational aspect.
From: George Greene on 18 Nov 2009 11:07
On Nov 18, 10:16 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > I guess I'm confused about the ontological status of abstract > objects such as sets. That's sort of unforgivable. The ontological status of ALL abstract objects IS JUST THAT -- "abstract object". > It seems silly to think of them realistically, as if there exists, > in some Platonic heaven, the universe of sets, That is ridiculous. YOU ALREADY SAID "abstract". There simply IS NO DIFFERENCE between "abstract" and "platonic". It seems ridiculous to pretend that the letter 'a' does not exist. |