Counterintuitions and the well-ordering theorem
in [Logic]
|
Prev: Solutions manual to Entrepreneurship 1e Bygrave Zacharakis
Next: Solutions manual for Engineering Mechanics Statics (12th Edition) by Russell C. Hibbeler
From: George Greene on 24 Nov 2009 10:33 On Nov 20, 9:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > If you introduce a class V of all sets, then we can > create a second object that is all subclasses of V, and another object > that is all the subclasses of the second object, etc. Then this whole > *super* universe can be reinterpreted as a theory of sets alone, where > V is reinterpreted not as *all* sets, but all sets of hereditary cardinality > less than some inaccessible cardinal alpha. Under this reinterpretation, > there *is* no object containing all of the (new) sets. This is the kind of thing that gives set theory a bad name. > So talk about completed totalities of sets can always be so reinterpreted > without losing anything important. If you do this, you have lost TWO VERY important things: 1) a class of all sets, since "under this reinterpretation, there is no object containing all of the (new) sets", AND 2) access to some of your cardinals.
From: George Greene on 24 Nov 2009 10:43 On Nov 23, 4:11 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > Yes, it does, but what I am doing is specifying a *partial* > axiomatisation of second-order logic and using that for my theories, > so that I have recursively enumerable theories. But the whole point of THAT is that you just might as well be doing FIRST-order logic. I seem to hear you saying that "full" 2nd-order logic is so intractable that when people SAY "2nd-order logic", by default, withOUT any such qualifier as "full", they CAN'T POSSIBLY mean 2nd-order logic, because that's just too hairy to even deal with. They ALWAYS (or at least by Default) mean some more tractrable first-order approximation. I'm just not in that linguistic community. Where I come from, it is actually important for 2nd-order logic TO MEAN 2nd-order logic, BY DEFAULT; the burden of qualifying adjectives rests PROPERLY on the people employing the truncated versions, most of which are a variation on the theme OF FIRST-order logic IN ANY case. There is a standard semantics for 2nd-order logic and it is Always the one with "full" powersets. The fact that there IS NO 1st-order way OF EVEN SAYING that you MEAN "full" powersets is THE REASON WHY "2nd-order logic" OUGHT to (by default) mean that.
From: George Greene on 24 Nov 2009 10:52 On Nov 23, 4:20 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > It is a claim that a certain second-order > sentence is not in fact satisfiable, even though ZFC says it is. "ZFC says it's satisfiable" means that ZFC *proves, as a theorem*, the existence of a model that makes the sentence true. That means that EVERY MODEL OF ZFC has SOME set in it that IS a model of the allegedly satisfiable sentence. But again, the order-confusion looms: how can FIRST-order ZFC ever even HOPE to SAY ANYTHING about ANY SECOND-order sentence?? You basically have to use 2nd-order ZFC to say anything meaningful about 2nd-order anything, and "2nd-order ZFC" is almost a contradiction in terms because if you actually understood the full-powerset semantics ALREADY, there wouldn't be much of anything left for ZFC to be investigating. You would already know whether the axiom of choice and the continuum hypothesis and the axiom of projective determinacy and a whole host of other things were vs. weren't "true", regardless of what was provable.
From: WM on 24 Nov 2009 12:12 On 24 Nov., 04:10, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Nov 21, 1:14 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > On 20 Nov., 13:37, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > Bill Taylor says... > > > > >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > >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. > > > > I don't see any big difference between the two. The set > > > of naturals and the set of reals are both abstractions. > > > I don't understand in what sense either exists, other > > > than exists as a coherent topic of study. > > > None of them does exist other than as a name and a wrong, i.e., self- > > contradictory idea. We can write sequences of symbols that allow us to > > talk about numbers and to manipulate numbers. That's all that exists - > > and it's enough to do mathematics. Everything else is a useless object > > for useless Fools Of Matheology. > > > Regards, WM > > That's not such an uplifting take on it. If you're really smart and > study really hard, then you can learn to write down symbols in ways > that a small handful of other people find interesting and worthy of > respect, but which just about everyone has no hope of understanding > and couldn't care less about, and fortunately the government is > prepared to subsidise it. > > We like to think that the story is more uplifting than that.- Zitierten Text ausblenden - Others like to use hashish. I don't recommend either of them. Regards, WM
From: Keith Ramsay on 29 Nov 2009 03:15
On Nov 22, 10:51 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: |Keith Ramsay says... [...] |>This is an issue for me with the axiom of choice. I've made a |>small stab at coming up with a satisfactory interpretation of |>set theory with the axiom of choice. Let's say, with a set of |>representatives for R/Q. Now I have a model that would be |>adequate, L, if all I wanted was to make the usual axioms and |>AC true in the model. But it interprets statements wrong. It |>converts the existence of a measurable cardinal into an odd |>and provably false statement about ordinals; but that's not |>what I mean by the existence of a measurable cardinal. It |>converts the existence of 0# into an odd and provably false |>statement, but one which doesn't mean what I mean if I were |>to talk about the existence of 0#. | |I'm not sure I understand this "conversion" process. Could |you say a little more about it? Please excuse the delayed reply. Conversion in the above example would simply be relativizing to L. The statement, "there exists a measurable cardinal" when relativized to L becomes a false statement, and likewise for the existence of 0#. But the meaning I have in mind for each statement is not the same (for one thing, neither is recognizably false on the grounds that their relativizations to L are recognizably false). [...] |>I think it's a valid concern that impredicative definitions might |>be consistent without being entirely meaningful. | |I just wanted to point out that impredicativity is only relevant |if you think that the objects are being "brought into existence" |by the definition. Perhaps this is usually the reason, but I don't think it has to be the reason. For me, impredicativity is mainly an issue of meaning. You can feel that you have a better grasp of what a statement means that quantifies over a domain if you have a grasp of what the domain is. You can feel that you have a better grasp of the nature of a domain of sets if you have a grasp of the meanings of the statements defining membership in an element of the domain. (What sort of claim will count as "n is in S"?) Being able to define your domains and/or statements in a well-ordering, so that domains are always defined in terms of prior types of statement, and types of statement in terms of quantification over prior domains, can make you feel better, therefore, even without any view of the domains and types of statement being "created" in a temporal sequence. We tend to think that it's okay to say, "use any property of the elements of a (given) S" as a defining property of a subset of S, but that's just sort of like saying "just max out on both sorts of things in the above". |If I say "Tom is the tallest person in the |room", I've identified Tom by a quantification over a set that |includes Tom as a member, so it's impredicative. However, that |use of impredicativity is harmless because Tom existed prior |to the definition. The definition is just selecting Tom, not |creating him. Mathematical impredicative definitions are similarly |harmless if you believe that the objects exist independently of |their definitions. Sure, but if you're playing the game of justifying the use of certain terms by supplying a model of their properties, already having a belief in the existence of the sorts of things being described is close to begging the question. If I believe already that there is such a thing as "the sets in the cumulative hierarchy", I must already have all the justification I think I need. Keith Ramsay |