The Definition of Points
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From: Lester Zick on 14 Apr 2007 15:30 On 14 Apr 2007 04:20:31 -0700, "Mike Kelly" <mikekellyuk(a)googlemail.com> wrote: >>I'm not the only one in the revolution against blind axiomatics. > >Blind axiomatics? So you think ZFC was developed by blindly? People >picked the axioms randomly without any real consideration for what the >consequences would be? Please. ZF(C) provides a foundation for >virtually all modern mathematics. This didn't happen by accident. Sure it did. Someone decided they wanted SOAP operas instead of mathematics and the rest is history. >What's "blind" about ZF(C)? What great insight do you think is missed >that you are going to provide, oh mighty revolutionary? What >mathematics can be done with your non-existant foundation that can't >be done in ZF(C)? Spelling apparently. ~v~~
From: Virgil on 14 Apr 2007 15:39 In article <462117d1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Blind axiomatics? So you think ZFC was developed by blindly? People > > picked the axioms randomly without any real consideration for what the > > consequences would be? Please. ZF(C) provides a foundation for > > virtually all modern mathematics. This didn't happen by accident. > > > > What's "blind" about ZF(C)? What great insight do you think is missed > > that you are going to provide, oh mighty revolutionary? What > > mathematics can be done with your non-existant foundation that can't > > be done in ZF(C)? > > > > 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? > >> Then how do you presume to declare that my statement is "not true"? > >> > > No answer? Do you retract the claim? > > >>> 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. > > OK so all of the above comes down to you demanding that we don't call > > cardinality "size". If we don't call cardinality "size" then all your > > objections to cardinality disappear. > > It would also be nice to have an alternative to cardinality Why? > > So, what's your opinion of infinite-case induction, IFR and N=S^L, and > multilevel logics, again? I forget. AS presented by TO, garbage, garage and garbage. There is a transfinite induction but it doesn't work the way TO would have it work, and there are a wide variety of logics none of which seems to work at all like TO would have them work.
From: Lester Zick on 14 Apr 2007 15:42 On 14 Apr 2007 04:20:31 -0700, "Mike Kelly" <mikekellyuk(a)googlemail.com> wrote: >> > Here's a hint, Tony : Ross and Lester are trolls. They don't beleive a >> > damn word they say. Naturally I don't have to believe a word I say when I can and do prove every word I say whereas mathematikers are required to believe every word they say and find faith essential because they can't. > They are jerks getting pleasure from intentionally >> > talking rubbish to solicit negative responses. Responding to them at >> > all is pointless. Responding to them as though their "ideas" are >> > serious and worthy of attention makes you look very, very silly. >> >> Yes, it's very silly to entertain fools, except when they are telling >> you the Earth is round. One needn't be all like that, Mike. When you >> argue with a fool, chances are he's doing the same. > >Nope. Ross and Lester are trolls. They are laughing at you when you >agree with their fake online personas. Continue wasting your time on >them if you like. I don't take Tony to be quite the fool I take you to be, Mike. At least Tony's capable of judging issues you prefer to avoid with hostility and vituperation so characteristic of those who prefer to judge people instead of issues. I've always found it interesting mathematikers convert so readily from numbers to psychology the moment their word count exceeds that of primitive people, you know "one, two, three, . . . many". ~v~~
From: Lester Zick on 14 Apr 2007 15:46 On 14 Apr 2007 00:09:21 -0700, "Brian Chandler" <imaginatorium(a)despammed.com> wrote: >What a good job then, that mathematics is not about "your >sensibilities". If you accept (use, adopt, whatever) the Axiom of >Choice then there is a proof that any set has a well-ordering. >Sensibilities don't come into it. I wonder though, Brian, if there is an axiom of forgiveness in modern math and how I might go about using, adopting, or whatever that axiom in my SOAP opera proofs? For that matter I wonder if there might also be an axiom of truth in modern math or are we just forced to wing it? ~v~~
From: Virgil on 14 Apr 2007 15:53
In article <46211c0a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <461fd938(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > > > >>> Nothing in TO's definition of "<" prohibits '(x>y) and (y>x)' from being > >>> true, so if he wishes to require such a prohibition, he must > >>> specifically add it to his transistivity requirement. > >>> > >> Yeah, actually, I misspoke, in a way. Your statement is still blatantly > >> false, in any case. It's possible for x<y and y<x in a cyclical-type > >> system, but those two facts together do not imply x=y. > > > > > > But a "cyclical-type system" is not an "ordered system" in any standard > > mathematical sense. > > Times of day have no order? Pulllease!!! Does 12 midnight on a clock come before or after 12 noon on that clock? There is an assumed local order in the sense that if two clock times are close enough together one usually assumes that one of them is "before" and the other "after", but is one minute before midnight clocktime before or after one minute after midnight clocktime? it could be either. > > > > > 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. So that I have better cause to be worried about TO than he has to worry about me. .... > >> The rationals are defined by NxN, minus the redundancies in > >> quantity within the matrix. > > > > That "matrix" is a geometric interpretation, which is quite irrelevant. > > > > A better definition for the rationals, based on I as the set of integers > > and P as the set of strictly positive integers is the set IxP modulo the > > "==" relation defined by (a,b) == (c,d) iff a*d = b*c. > > > > > > > >> Equinumerous to those redundancies, which > >> are the vast majority of cells, are the irrationals. That's how it > >> actually works. > > > > 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? |