The Definition of Points
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From: Tony Orlow on 18 Apr 2007 14:55 Lester Zick wrote: > On Tue, 17 Apr 2007 13:33:39 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>>> Is there a set >>>> of statements S such that forall seS s=true? >>> No idea, Tony. There looks to be a typo above so I'm not sure exactly >>> what you're asking. > >> I am asking, in English, whether there is a set of all true statements. > > No. There are predicates to which all true statements and all false > statements are subject respectively but no otherwise exhaustively > definable set of all true or false statements because the difference > between predicates and predicate combinations in true or false > statements is subject to indefinite subdivision. > > ~v~~ Uh, what? 01oo
From: Tony Orlow on 18 Apr 2007 14:57 Lester Zick wrote: > On Tue, 17 Apr 2007 13:29:44 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> Let me ask you something, Tony. When you send off for some truth value >>> according to "true(x)" and it returns a 1 or 0 or whatever, how is the >>> determination of that "truth value" made? >> From the truth values of the posited assumptions, of course, just like >> yours. > > So you just posit truth values and wing it whereas I'm more inclined > to demonstrate the truth of what I posit instead? > > ~v~~ What truth have you demonstrated without positing first? 01oo
From: Tony Orlow on 18 Apr 2007 15:04 Mike Kelly wrote: > On 18 Apr, 06:17, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >> >> < snippery > >> >>> You've lost me again. A bad analogy is like a diagonal frog. >>> -- >>> mike. >> Transfinite cardinality makes very nice equivalence classes based almost >> solely on 'e', > > Why "almost solely"? > The successor relation is a relation not related to 'e' except in a particular model of the naturals upo which the rest is built. >> but in my opinion doesn't produce believable results. > > Which "results" of cardinality are not believable? You don't think the > evens and the naturals and the rationals are mutually bijectible? Or > you do think the reals and the naturals are bijectible? Or did you > mean you have a problem not with what cardinality actually is but with > what you hallucinate cardinality to be? > As long as transfinite "equivalence" classes are referred to as such, and not said to denote "equal" size or numerosity, then I have no problem. >> I'm working on a better theory, bit by bit. > > You have never managed to articulate a problem with what cardinality > actually says (all it says is which sets are bijectbile). You've never > managed to explain the motivation behind your "better theory". I don't > even know whether your ideas are supposed to be formulated in ZF or in > something else, possibly a new foundation of your own devising(or, > perhaps, never formalised at all?). FWIW I don't think it would be > very hard to define, say, "density in the naturals" in ZF. I just > don't see the motivation behind doing so. > The idea is to create a method that works for comparing countable and uncountable sets alike, and properly defining the relationship between the two. >> I think trying to base >> everything on 'e' is a mistake, since no infinite set can be defined >> without some form of '<'. I think the two need to be introduced together. > > But you are, in fact, completely mistaken here. Several people have > told you this. You have some inability to acknowledge this. Oh well. > > PS you snipped a lot of my post; maybe you found it boring. I find > your evasiveness quite interesting, though. What does cardinality > claim to tell that it can't? > > -- > mike. > I suppose it's more the way it's spoken about. Leave out "size" and "equinumerous" and talk about "equivalence classes" and not "equality", and I have no problem with the consistency of standard theory. tony.
From: MoeBlee on 18 Apr 2007 15:08 On Apr 17, 9:56 pm, Tony Orlow <t...(a)lightlink.com> wrote: > MoeBlee wrote: > > On Apr 17, 11:33 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> But, I have a question. What, exactly, is the difference between > >> "equality" and "equivalence"? > > > As I told you, in first order, we cannot generalize (I'll say about > > exceptions in a moment) that identity can be "captured" by axioms. > > Thus in the general case, we require stipulating a fixed semantics for > > the identity symbol. Meanwhile, equivalence often pertains to an > > equivalence relation (reflexive, symmetric, transitive). Every > > equiavlance relation on set induces a partition of the set; that > > partition being a set of equivalence classes. Then two objects are > > equivalent iff they are both members of the same equivalence class, > > which is to say that they bear the equivalence relation (the > > reflexive, symmetrc, transitive relation) to each other. Notice that > > identity is a an equivalence relation but not all equivalence > > relations are the identity relation. > > Okay, distinction noted. It would seem to apply well to cardinality, > where we have classes of sets that match, even if they are not equal as > sets, eh? I wouldn't put it that way, but I think you're onto the basic idea. Yes, bijectability (equinumeroisity) is an equivalance relation on any set. > >> Is it not, in the tradition of Leibniz, a > >> matter of detecting a distinction between two objects, or not? That is, > >> if we can detect no difference between two objects, if we can find no > >> attribute which distinguishes them, then are they not "equal" or > >> "equivalent", or "identical"? Ultimately, if we can not say, in one > >> sense or another, that x<y or y<x, then do we not consider that x=y? > > > One direction of Leibniz's principle (the indiscerniblity of > > identicals) is acheived by an axiom that if x = y then whatever holds > > for x holds for y. But, in first order, we cannot generalize that the > > other direction (the identity of indiscernbiles) can be stated even as > > an axiom schema. Thus a stipulated semantics is given for the identity > > symbol. > > And this is because first order logic does not allow the universal > quantifier applied to sets or properties, but only elements, is that > right? Again, I wouldn't put it that way, but you're on to the right idea. In first order, we don't quantify over predicates (we don't follow the universal or existential quantifier by a variable that ranges over sets of n-tuples (n>0) of members of the domain of discourse, since we have no such variables). > Is it not possible to concoct a logic where the first order of > analysis is on the property, versus object, level? I'll have to dream > about that a bit...what do you think? I don't know what the point would be. If you are going to quantify over predicates, then why not just quantify over both predicates and individuals, which is what second order logic is. > >> Defining "=" depends, as far as I can tell, on defining "<". Is this > >> wrong, in your "opinion"? > > > In what context? In set theory, given an appropriate axiomatization, > > we can define '=' from just 'e' (the membership symbol). By the way, > > that does not contradict my remarks about first order not being able > > to "capture" identity, since what that says is that it is not the case > > that for ANY langauge or theory we can "capture" identity. For CERTAIN > > languages and theories (viz. those with only finitely many predicate > > symbols) we can "capture" so as to make a definition for the identity > > symbol that conforms to the basic semantics that 'x=y' is true iff > > denotation of x and the denotation of y are identical. > In a very real sense, for all xeX, x<X, so defining '=' from 'e' is > pretty similar to defining it from '<'. That's the definition of '<' for ordinals. It's not for sets in general. And it wouldn't work for equality of sets in general as long as we have regularity, since it is never the case that xey & yex. The actual definition can be made by the subset relation: x=y <-> (x subset_of y & y subset_of x). > I understand that this is not > pure first order logic, but first order with an added operator with its > own properties. First order by itself can't distinguish among anything > but truth values, I'm not sure what you mean by that. > so if those pertain to some objects or properties, the > additional relationships between those need to be handled with operators > that take as parameters whatever type of objects or properties one has, > rather than just truth values. In the context of whatever additional > operators are added to first order logic, indeed, if every statement > true of each is true of the other, then those two objects or properties > are considered identical. In set theory, since there are only finitely many predicate symbols, we CAN and we DO implement both directions of Leibniz's principle. > That's not to say that an additional operator > might not produce a difference between two previous equated elements, is it? I don't follow what you meant about operators. But why don't you get 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar? I'm not saying that challengingly (this time). I am sincerely suggesting that such books would give you start toward a solid basis upon which to develop and communicate your own ideas and possible innovations with things such as operators. MoeBlee
From: Tony Orlow on 18 Apr 2007 15:13
MoeBlee wrote: > On Apr 17, 10:02 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> I am saying it's obvious that any countable set has a well ordering. It >> is not obvious for uncountable sets. > > It's not only obvious, but it's trivial to prove that every countable > set has a well ordering. > > So, just to be sure you understand, what is at stake with an axiom of > countable choice is not at all that every countable set has a well > ordering (since we don't need any choice principle to prove that) but > rather that for any countable set there is a function that chooses > exactly one member from every nonempty subset of that countable set > (and actually, we need concern ourselves only with denumerable sets, > since we don't need a choice principle to prove that for every finite > set there is a function that chooses exactly one member from every > nonempty subset of the finite set). > > MoeBlee > It's not trivial to prove any such thing for an uncountable set. That requires full Choice, does it not? In fact, it was Virgil, I believe, who said that was the MOTIVATION behind Zermelo's formulation of Choice, so that ALL sets could be considered well orderable. However, while I see that a choice function produces a well ordering for any countable set, I don't see that it does for an uncountable set. Maybe I am confused about something. Can a well ordering include an uncountable number of limit elements? If so, then I can see how such a well ordering could occur. Are limit elements which are an infinite number of limit elements beyond the first what are called "inaccessible" limit ordinals? If this is the case, then I guess I can see that in this sense, one could concoct a well ordering. But, no, then there would exist an infinite descending sequence of limit ordinals withoin the well order. So, that wouldn't work, right. I just don't see that a well order can be accomplished in principle on an uncountable set, whether one declares an axiom to that effect of not. TOEknee |