The Definition of Points
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From: Virgil on 20 Apr 2007 14:56 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.
From: Virgil on 20 Apr 2007 14:59 In article <4628e340(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > i is a number produced by extending completeness to the multiplicative > field, after introducing completeness to the additive field to produce > -1, like I said above. You do not have a field if either addition or multiplication is 'incomplete'.
From: Virgil on 20 Apr 2007 15:05 In article <4628e6bd(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Just take w > > +1. The well ordering on w+1 by the membership relation has w itself > > as member of the field of the ordering. But to "travel from" w "to" > > the least member of w requires "travelling through" an infinte number > > of members of the domain of the field of the ordering. > > As it was explained to me, because every natural is finitely far from > the lest in w, there is no infinite descending chain of predecessors one > can define. In other words, between any two limit ordinals can only a > countable number of elements. Irrelevant. The point here is that in any well ordered set, between any limit member and any prior member there are infinitely many other members.
From: Tony Orlow on 20 Apr 2007 15:09 MoeBlee wrote: > On Apr 20, 9:24 am, Tony Orlow <t...(a)lightlink.com> wrote: >> MoeBlee wrote: > >> So, defining N doesn't involve a successor relation between two >> elements, as well as a member relation between an element and a set? > > The successor operation is DEFINED in terms of the membership > relation. EVERYTHING in set theory is defined in terms of the > membership relation. The only non-logical primitives of set theory are > '=' and 'e' (and we could even define '=' in terms of 'e' if we want > to set it up that way). There is NO formula of set theory that doesn't > revert to a formula in the primitive language with just 'e' and > '=' (or even just 'e') as the ONLY non-logical symbols. I've been > telling you this for probably over a year now. Why don't you > understand this? > The particular successor relation for the vN ordinals is defined in terms of 'e', but that's not the only model of successorship. Successors don't necessarily have to be supersets. They are usually "greater" in some sense, in which case subsets thereof can be formulaically compared for infinite size, as long as the mapping preserves the same overall '<' relationship between successive elements. You remember my beef about not-quantitatively ordered mappings, don't you? >> When you define N, doesn't the rule E x e N -> E succ(x) e N define a >> relation between two elements, as well as between those elements and N? > > "Define a relation". From N of course we can define the successor > relation on N. So what? That doesn't refute that every definition in > set theory ultimately reverts to the membership relation. I'll say it > YET AGAIN: Before you go on your rant, let me just say this. As a relation, I specifically mean that each combination of inputs produces either a 0 or 1. It's a truth table conception of what a relation is. With successor, the inputs are each a member of N, a natural, and certainly only one cell in each row and one cell in each column will be a 1. That's a relation from each natural to each other, only one of which is its successor. There is also the relation of membership, which each of these ordinals has with w. That is a separate relation. Each such natural we define is a member of N, so we have 1's in the w column for all of them. If we define other columns for other sets, like R, and include all their members, then we will create new members, with 0's in the w column. So, successor(x,y) and element(x,y) are two different relations, one of order, and one of membership. But let's see what you say... > > The only non-logical primitives of set theory are '=' and 'e' (and we > could even define '=' in terms of 'e' if we want to set it up that > way). There is NO formula of set theory that doesn't revert to a > formula in the primitive language with just 'e' and '=' (or even just > 'e') as the ONLY non-logical symbols. Is '=' non-logical? It seems to me: equals(a,b) = and(implies(a,b),implies(b,a)) '=' <> '<->' ? But of course, '=' means not '<>', but those are different arrows. > > Over a year I've been telling you that over and over and over. But it > seems that as far as you're concerned, such information is just a > random collection of characters appearing on a computer monitor. > > MoeBlee > > Repeating the same thing doesn't usually do much to change the conversation. >> TOEknee
From: Virgil on 20 Apr 2007 15:10
In article <4628e964(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > There have been lots of objections, and a few valid points, but no major > flaws detected in what I propose. It's just not compatible with ZFC. TO simply chooses to ignore the many major flaws that have been pointed out to him in such miniscule snippets of his various mutually incompatible systems that have been presented here. > > >>> Get this through your head : every relation between objects in set > >>> theory is based on 'e'. It's really pathetic to keep mindlessly > >>> denying this. Set theory doesn't just "try to base everything on 'e'". > >>> It succeeds. > > > >> If you say so. > > > > No, not just "if he says so". He says so and he's RIGHT. And you can > > VERIFY for yourself just by reading a textbook already. > > > > MoeBlee > > > > So, defining N doesn't involve a successor relation between two > elements, as well as a member relation between an element and a set? All that is done in terms of 'e', so 'e' is 'e'nough to get N. |