The Definition of Points
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From: MoeBlee on 23 Apr 2007 15:00 On Apr 23, 10:29 am, Tony Orlow <t...(a)lightlink.com> wrote: > Virgil wrote: > > Successor, in the x -> x u {x} sense, depends only on singleton sets, > > subsets and unions, all of which are defined strictly in terms of 'e', > > so what is left? Nothing. > > Only the implication from the existence of one element to the next, > which defines the successor relation as one between two elements, and in > this case, equivalent to 'e'. If we define successor in the x -> x+1 > sense, then it's not really based on 'e', but it does define the > naturals, in a quantitative way. Successor CAN be related to 'e', but > really depends on recursive implication of existence. "recursive implication of existence." Oh, brother! MoeBlee
From: Virgil on 23 Apr 2007 16:14 In article <462ceab3(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > But what your 1. and 2. don't take into account is that we can (and > > "frequently" must have), given any two elelemnt x,y in S: > > > > S > > / \ > > / \ > > > > S-{x} S-{y} > > > > \ / > > \ / > > > > S-{x,y} > > > > > > So the "tree" criss-crosses like a chain link fence. This type of > > partial order is usually called a lattice. > > Yes, it becomes a sort of a lattice-looking thing from one level to the > next. It's actually the set of vertices of a |S|-dimensional cube, if > the same subset may only occur once. If you allow the redundancies, so > that S-[x,y] appears both as a child of S-{x} and of S-{y}, then you get > a tree, but not every element is unique. I guess on each level n, where > S is on level 0, one gets each unique subset n times, and the number of > unique elements generated at each level is 2^n-n? Something like that. In a tree structure, as defined in mathematics, no 'child' can have more than one 'parent'. What you are describing should not be called a tree, but perhaps you mean a lattice, in which a 'child' can have any number of patents, like in the lattice of subsets of a given set with the relation of 'subset of' as the 'child' relation.
From: Virgil on 23 Apr 2007 16:19 In article <462cecec(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4628df20$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >>> I do say so. Does this mean you're going to stop claiming that > >>> relations in set theory aren't based solely on 'e'? > >>> > >>> -- > >>> mike. > >>> > >> No, not when sequences are defined using a recursively defined successor > >> function, which is a relation between two elements, as opposed 'e', a > >> relation between an element and a set. The combination of the two is > >> what produces an infinite set, no? > > > > So that x -> x u {x} does not depend on 'e' ? > > > > On can define successor entirely in terms of 'e'. > > > > Successor, in the x -> x u {x} sense, depends only on singleton sets, > > subsets and unions, all of which are defined strictly in terms of 'e', > > so what is left? Nothing. > > Only the implication from the existence of one element to the next, > which defines the successor relation as one between two elements, and in > this case, equivalent to 'e'. If we define successor in the x -> x+1 > sense, then it's not really based on 'e' T here is a reasonable way of defining x+1 in tems of 'e', but I have never seen it done without ever using 'e'. Perhaps TO can provide an axiom system allowing 'x+1' for each 'x' but not ever relying on any sets or any membership, so that we can see what it looks like? > but it does define the > naturals, in a quantitative way. Successor CAN be related to 'e', but > really depends on recursive implication of existence. Which itself depends on 'e'.
From: Ben newsam on 23 Apr 2007 19:43 On Mon, 23 Apr 2007 11:54:05 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >If you want to talk about the truth values of individual facts used in >deduction, by all means, go for it. I would counsel you seriously not to attempt it, Lester will only obfuscate the "discussion", and then will start to hurl personal insults at you.
From: Ben newsam on 23 Apr 2007 19:54
On Mon, 23 Apr 2007 11:47:26 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: >On Mon, 23 Apr 2007 11:54:05 -0400, Tony Orlow <tony(a)lightlink.com> >wrote: >> Truth tables and logical statements involving variables are >>just that. If I say, 3x+3=15, is that true? No, we say that IF that's >>true, THEN we can deduce that x=4. > >But here you're just appealing to syllogistic inference and truisms >because your statement is incomplete. You can't say what the "truth" >of the statements is or isn't until x is specified. So you abate the >issue until x is specified and denote the statement as problematic. > >The difficulty with syllogistic inference and the truism is Aristotle >never got beyond it by being able to demonstrate what if anything >conceptual was actually true. The best he could do was regress >demonstrations of truth to perceptual foundations which most people >considered true, even if they aren't absolutely true. But even based >on that problematic assumption he could still never demonstrate the >conceptual truth of anything beyond the perceptual level. Here we go, tap dancing in porridge again. |