The Definition of Points
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From: MoeBlee on 18 Apr 2007 14:16 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
From: Tony Orlow on 18 Apr 2007 14:16 Virgil wrote: > In article <462507cf(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <462117d1(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> Axiomatically, I think the bulk of the burden lies on Choice in its full >>>> form. Dependent or Countable Choice seem reasonable, but a blanket >>>> statement for all sets seems unjustified. >>> Since it has been shown that if ZF is consistent then ZFC must be >>> consistent as well, what part of ZF does TO object to? >>> >> Well, I also put the onus on the Extensionality, as far as equating sets >> with the same general membership, but different rates of growth per >> iteration, but I haven't quite figured out how to formalize that >> statement, or at least, am not in a position to do so now. > > I.e., vaporware! >>> >>>>>> Then how do you presume to declare that my statement is "not true"? >>>>>> >>>> No answer? Do you retract the claim? >>>> >> Yes? >> >>>>>>> It's very easily provable that if "size" means "cardinality" that N >>>>>>> has "size" aleph_0 but no largest element. You aren't actually >>>>>>> questioning this, are you? >>>>>> No, have your system of cardinality, but don't pretend it can tell >>>>>> things it can't. Cardinality is size for finite sets. For infinite sets >>>>>> it's only some broad classification. >>> It is one form of size for all sets. One might use the physical analogy >>> that volume, surface area, and maximum linear dimension are all measures >>> of the size of a solid. So implying that one "size" fits all is false. >>> >>> >> Each of those is derivative of the last, given the proper unit of >> measure. > > So does one use distance, area, volume? > > What is the maximum length of a solid in cubic meters, or in square > meters? > What is its surface area in cubic meters or in linear meters? What drug are you on? Do you know any physics? Distance is in terms of d, say. Then area is in terms of d^2 and volume in terms of d^3, etc. So, it stands to reason that, since the derivative of a polynomial is one of the next lower power with respect to whichever variable one is considering, the x-dimensional boundary of a x+1-dimensional object be the derivative of the contained space. With the circles, it's clear. The volume of a sphere is 4/3 pi r^3, and the surface area its derivative, 4 pi r^2. The circumference of a circle is 2 pi r, whose integral, pi r^2, is the area contained. In this case, r appears to be the proper unit of measure to make these equations work. In a regular polygon or polyhedron, one might wonder which r to use. As it turns out, r is from the center of the object to the center of each face or side, and not to the vertices. Using that distance as one's unit of measure always makes the boundary equal to the derivative of the contained space. One of these days it might be nice to generalize this to irregular polygons and polyhedra. I have a feeling it might be a simple matter to determine the proper unit given the set of points, but then, it might get hairy.
From: Tony Orlow on 18 Apr 2007 14:18 Virgil wrote: > In article <46250ad6(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >> Yes, it can be that x<y and y<x and y<>x. > > Then that is not an order relation. In any form of order relation "<" > allowed in mathematics (x<y and y<x) requires x=y. >>>>> For any in which "<" is to represent the mathematical notion of an order >>>>> relation one will always have >>>>> ((x<y) and (y<x)) implies (x = y) >>>>> >>>> Okay, I'm worried about you. You repeated the same erroneous statement. >>>> You didn't cut and paste without reading, did you? Don't you mean "<=" >>>> rather than "<". The statement "x<y and y<y" can only be true in two >>>> unrelated meanings of "<", or else "=" doesn't have usable meaning. >>> TO betrays his lack of understanding of material implication in logic. >>> For "<" being any strict order relation, "(x<y) and (y<x)" must always >>> be false so that any implication with "(x<y) and (y<x)" as antecedent >>> for such a relation, regardless of conseqeunt, is always true. >> Oh, yes, well. Any false statement implies any statement, true or false, >> as long as you're not an intuitionist. If (x<y) -> ~ (y<x), then x<y ^ >> y<x is of the form P ^ ~P, or ~(P v ~P) which is false in classical >> logic, but not intuitionistically. There is debate on this topic. > > I very much doubt that any intuitionist would say that for an order > relation,"<", on any set one could have x<y and y<x without having x=y. > >>>>> Is TO actually claiming that the irrationals form a subset of the >>>>> countable set NxN. >>>>> >>>>> That is NOT how it works in any standard mathematics. >>> Then why does TO claim it? >> I am not. I am saying it is a set equal in magnitude to the redundancies >> in NxN. > > The set of "redundancies in NxN" must be a subset of NxN itself, so must > be countable, so that TO is saying that the set of irrationals is "equal > in magnitude" to a countable set. Well, actually, I mean N*xN*, including infinite naturals in N*. Sorry, I should have been more specific.
From: Tony Orlow on 18 Apr 2007 14:18 Virgil wrote: > In article <46250ea8(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> In all fairness to Lester > > Why bother to be fair to one who is so compulsively unfair? To set a good example?
From: Tony Orlow on 18 Apr 2007 14:24
Virgil wrote: > In article <46251314(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> But, I have a question. What, exactly, is the difference between >> "equality" and "equivalence"? > > "Equality" usually means being the same in every detectable respect. > "equivalent" usually means the same in one respect, or in a limited > number of respects, while still possibly distinguishable in other > respects. > > > For example, parity of a natural number is an equivalence. > > Naturals having the same parity (evenness or oddness) are equivalent > with respect to parity even though not equal as natural numbers. > > 1 and 3 are equivalent with respect to parity but are not equal. Right. I stand corrected. Thanks you. But, the question remains in this regard. How do we know we have actually looked at all possible respects in which could compare? When we know of some way in which two things differ, but under a limited set of criteria they are not distinguishable, I can see that they are "equivalent" in that "respect". But, how can we be sure that we have actually taken into account all respects in which two objects may be compared? For instance, there seems to be some who consider "0.999...=1" somewhat suspect. Could we not consider two slightly different interpretations for these strings, considering them to be "equivalent" as standard reals, while not being exactly "equal" in every respect? |