The Definition of Points
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From: Lester Zick on 17 Apr 2007 19:30 On Tue, 17 Apr 2007 13:29:44 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >> So what all this nonsense comes down to is that your "truth values" >> have no beginning in actual mechanical terms because they're given to >> you by assumption and not demonstrated in mechanical terms and all >> those conjunctions and conjunctive manipulations you describe are just >> so many arbitrary translation rules to work with otherwise meaningless >> 1's and 0's. >> >> ~v~~ > >0 and 1 are meaningful. They are nothing and all. Not interested, Tony. Save it for church. ~v~~
From: stephen on 17 Apr 2007 20:21 In sci.math David R Tribble <david(a)tribble.com> wrote: > Tony Orlow writes: >>> ala L'Hospital's theft from the Bernoullis, and >>> the division by 0 proscription. >> > Alan Smaill wrote: >> and Zick was the one who claimed that he would use l'Hospital to work >> out the right answer for 0/0. such a japester, eh? > Geez. How many posts before someone points > out that it's l'H�pital's rule? But that would spoil the joke. Stephen
From: Brian Chandler on 17 Apr 2007 23:40 David R Tribble wrote: > Tony Orlow writes: > >> ala L'Hospital's theft from the Bernoullis, and > >> the division by 0 proscription. > > > > Alan Smaill wrote: > > and Zick was the one who claimed that he would use l'Hospital to work > > out the right answer for 0/0. such a japester, eh? > > Geez. How many posts before someone points > out that it's l'Hôpital's rule? L'Hospital is where > you take someone after they get punched in the > nose by a mathematician after saying "l'Hospital's rule". Well, given that "ô" is only a French spelling for "o-with-the- following-s-omitted", seems to me that l'Hospital is a pretty reasonable asciification. Brian Chandler http://imaginatorium.org
From: Tony Orlow on 18 Apr 2007 00:56 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? >> 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? 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 am totally prepared to admit that your understanding is correct and both >>>> Wolfram and Wikipedia are badly worded. There is lots of misinformation on >>>> the internet. What troubles me is that the books on set theory I have also >>>> have their definitions worded like wolfram and Wikipedia. >>> I'd suggest you check those books to see how they define "=". If they >>> don't define it at all, I'd put my money on the intended meaning being >>> identity. >> 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. > > MoeBlee > In a very real sense, for all xeX, x<X, so defining '=' from 'e' is pretty similar to defining it from '<'. 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, 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. That's not to say that an additional operator might not produce a difference between two previous equated elements, is it? TOEknee
From: Tony Orlow on 18 Apr 2007 01:17
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', but in my opinion doesn't produce believable results. So, I'm working on a better theory, bit by bit. 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. tony. |