The Definition of Points
in [Logic]
|
Prev: On Ultrafinitism
Next: Modal logic example
From: Tony Orlow on 31 Mar 2007 18:37 Virgil wrote: > In article <460e5198(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <1175275431.897052.225580(a)y80g2000hsf.googlegroups.com>, >>> "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >>> >>>> On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote: >>>> >>>>> They >>>>> introduce the von Neumann ordinals defined solely by set inclusion, >>>> By membership, not inclusion. >>> By both. Every vN natural is simultaneously a member of and subset of >>> all succeeding naturals. >>> >> Yes, you're both right. Each of the vN ordinals includes as a subset >> each previous ordinal, and is a member of the set of all ordinals. > > In ZF and in NBG, there is no such thing as a set of all ordinals. > In NBG there may be a class of all ordinals, but in ZF, not even that. > > No, that's true, The ordinals don't make a set. They're more like a mob, or an exclusive club with very boring members, that forget what their picket signs say, and start chanting slogans from the 60's. > >> Anyway, my point is that the recursive nature of the definition of the >> "set" introduces a notion of order which is not present in the mere idea >> of membership. Order is defined by x<y ^ y<z -> x<z. > > That is only a partial ordering on sets, and on ordinals is no more than > the weak order relation induced by membership, namely: > x < y if and only if either x e y or x = y. > > Yes, that's weak, but it's order. > > This is generally >> interpreted as pertaining to real numbers or some subset thereof, but if >> you interpret '<' as "subset of", then the same rule holds. I suppose >> this is one reason why I think a proper subset should ALWAYS be >> considered a lesser set than its proper superset. It's less than the >> superset by the very mechanics of what "less than" means. > > But your version of "less than" is is only a partial order > Oh no :o! How can I ever face my family again? >>> On the other hand, Tony Orlow is considerably less of an ignoramus than >>> Lester Zick. >> Why, thank you, Virgil. That's the nicest thing you've ever said to me. > > You would not think so if you knew my opinion of Zick. I didn't mean to imply it was actually nice.... ;) haha Thanks anyway. Tony
From: Bob Kolker on 31 Mar 2007 18:42 Tony Orlow wrote:>> > > Measure makes physics possible. On compact sets which must have infinite cardinality. The measure of a dense countable set is zero. Bob Kolker
From: Tony Orlow on 31 Mar 2007 18:45 Virgil wrote: > In article <460e5476(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <460d4813(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>> >>>> An actually infinite sequence is one where there exist two elements, one >>>> of which is an infinite number of elements beyond the other. >>> >>> Not in any standard mathematics. >> Well, "actually infinite" isn't a defined term in standard mathematics. > > If it is outside the pale, why bother with it at all? The pail is fine for minnows and tadpoles. Us crabs are always climbing out of the pail. Ah! I got your toe again! Nyah! >>> In standard mathematics, an infinite sequence is no more than a function >>> whose domain is the set of naturals, no two of which are more that >>> finitely different. The codmain of such a function need not have any >>> particular structure at all. >> That's a countably infinite sequence. Standard mathematics doesn't allow >> for uncountable sequences like the adics or T-riffics, because it's been >> politically agreed upon that we skirt that issue and leave it to the >> clerics. > > What sect would have been so foolish as to have ordained TO into its > priesthood? > The one that allowed me to circumcise you up to the elbow. :) > > However, where every element of a set has a well defined >> successor and predecessor, it's a sequence of some sort. > > Every such "sequence" set must be representable as a function from the > from the integers to that set. Why? What have I defined, if not a sequence? Is there a word for it? It must "exist", if I assert so. Thanks. Tony
From: Bob Kolker on 31 Mar 2007 18:47 Tony Orlow wrote: > > No, Bob, that's a Muslim lie, perpetrated by the Jews as a joke on the > xtians. Are you aware the Cantor was a Lutheran? Probably not. His arch enemy Kroeniker was Jewish. Bob Kolker
From: Tony Orlow on 31 Mar 2007 18:48
Lester Zick wrote: > On Fri, 30 Mar 2007 11:50:10 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> And the only way we can address >>> relations between zeroes and in-finites is through L'Hospital's rule >>> where derivatives are not zero or in-finite. And all I see you doing >>> is sketching a series of rules you imagine are obeyed by some of the >>> things you talk about without however integrating them mechanically >>> with others of the things you and others talk about. It really doesn't >>> matter whether you put them within the interval 0-1 instead of at the >>> end of the number line if there are conflicting mechanical properties >>> preventing them from lying together on any straight line segment. >>> >>> ~v~~ >> Well, if you actually paid attention to any of my ideas, you'd see they >> are indeed mostly mechanically related to each other, but you don't seem >> interested in discussing the possibly useful mechanics employed therein. > > Only because you don't seem interested in discussing the mechanics on > which the possibly useful mechanics employed therein are based, Tony. But I am. I've asked that you fill in those true/false entries in the table I gave you, so we can see what relation you're employing. That's an effort in "discussing" the "mechanics". > I'm less interested in discussing one "possibly useful mechanics" over > another when there is no demonstrable mechanical basis for the > "possibly useful mechanics" to begin with. You claim they're "mostly > mechanically" related but not the mechanics through which they're > "mostly mechanically related" except various ambiguous claims per say. > > ~v~~ Pro say, to be exact. How many inputs, how many outputs, and what mapping, what relation? Them's mechanics. So, expliculate. 01oo |