An uncountable countable set
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From: Lester Zick on 16 Oct 2006 15:29 On Sun, 15 Oct 2006 12:38:50 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > >> I find the idea absurd that natural numbers can be built >> by putting curly braces around the empty set. >> >> Han de Bruijn > >It appears as if much of useful mathematics is ultimately based on what >HdB finds absurd. Which only means mathemagicians find the absurd useful. ~v~~
From: Lester Zick on 16 Oct 2006 15:37 On 16 Oct 2006 08:07:56 -0700, imaginatorium(a)despammed.com wrote: > >Han de Bruijn wrote: >> Virgil wrote: >> >> > In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, >> > Han.deBruijn(a)DTO.TUDelft.NL wrote: >> > >> >> I find the idea absurd that natural numbers can be built >> >>by putting curly braces around the empty set. >> > >> > It appears as if much of useful mathematics is ultimately based on what >> > HdB finds absurd. >> >> The natural numbers can be defined without employing set theory. > >I should think they could be. Though I fancy the natural numbers will >never be properly defined by anyone who is incapable of understanding >the set theoretic definition. Well, Brian, if by "incapable of understanding . . ." you mean "unwilling to agree with set theoretic assumptions regarding definition of the naturals" I would certainly have to disagree. The naturals can certainly be defined without set theoretic assumptions. ~v~~
From: Lester Zick on 16 Oct 2006 15:38 On Mon, 16 Oct 2006 09:45:35 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Virgil wrote: > >> In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, >> Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >>> I find the idea absurd that natural numbers can be built >>>by putting curly braces around the empty set. >> >> It appears as if much of useful mathematics is ultimately based on what >> HdB finds absurd. > >The natural numbers can be defined without employing set theory. True. ~v~~
From: David Marcus on 16 Oct 2006 15:40 Ross A. Finlayson wrote: > Virgil wrote: > > In article <4530434f(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > David Marcus wrote: > > > > Han de Bruijn wrote: > > > >> Virgil wrote: > > > >> > > > >>> In article <9020$452f46c4$82a1e228$31963(a)news2.tudelft.nl>, > > > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>> > > > >>>> Virgil wrote about the Balls in a Vase problem: > > > >>>> > > > >>>>> Everything takes place before noon, so that by noon, it is all over and > > > >>>>> done with. > > > >>>> Noon is never reached, because your concept of time is a fake. > > > >>> > > > >>> No one expects the experiment to take place anywhere except in the > > > >>> imagination, so that everything about it, including its time, is > > > >>> imaginary, but logic continues to hold even there, at least for > > > >>> mathematicians. And logic says that a ball removed from a vase is not > > > >>> later in the vase. > > > >> Since your logic and the logic of others give contradictory results for > > > >> the same problem, logic alone is unreliable. > > > > > > > > Are you saying that Mathematics gives contradictory results for a > > > > problem? If so, please state the problem. > > > > > > > > > > The problem, among others, is the vase. If you haven't gotten a clue > > > about the problem yet, well, get on the bus. > > > > Along with TO, HdB, "Mueckenh", Ross, and others of that ilk. > > > > I am surprised that JSH has not joined in. > > There is no universe in ZF, ZF is inconsistent. What exactly is the inconsistency, please? If the following text that you wrote is supposed to be an answer to this question, I can't make it out. Please be specific and go one step at a time. > Model theory posits the existence of a maximal ordinal in, for example, > ZF, where there exists no maximal ordinal. That the maximal ordinal > does and does not exist is self-contradictory, in ZF and similar > regular/well-founded theories. The generic extension of N, that > bijects to R, contains no elements not in N. Sets are defined by their > elements. > > For no finite differential, as algebraically manipulable as it is, does > analysis work. The differential is infinitesimal. > > There are only real numbers between zero and one. There are also > everywhere real numbers between zero and one. If infinitesimals exist > between zero and one, they're reals. The least positive real is called > iota, and according to "Counterexamples in Real Analysis" such a thing > exists, arithmetic on those things involves book-keeping of related > rates and an implicit universe. There are a variety of developments of > "non-standard" real numbers, and the hyperintegers are the standard > integers and the hyperreals are the standard reals, because the > transfer principle is what's of actual importance in extending various, > but not all, results true on the finite to the infinite. > > A variety of modern mathematical methods divide by zero. > > A (regular) set, even an infinite set, is demonstrably larger than a > proper superset. > > ZF has no numbers in it. In general finite von Neumann ordinals are > considered to be mechanistically the natural numbers, but that is a > definition. Cauchy/Dedekind is insufficient to describe the real > numbers. > > N E N, in terms of the naturals number similarly to how U E U, the > universe would contain itself, besides that the universe as a set is an > element of itself. Where N is considered in a similar vein, because > it's infinite and Spinoza's continuum, N < N besides N = N. The > integers are superarchimedean ring. > > Less than absolute zero is hotter than any finite temperature. The > closer the subatomic particles are examined, the smaller they appear to > be, as reflected in CODATA. Similarly, the more that is known about > the physical universe, the larger it appears to be, it's infinite. > > There's only one theory with no non-logical axioms, the null axiom > theory. If it's not the null axiom theory, Goedel says it's not a > theory of everything. > > I'm among many who see utility in bijecting the naturals to the unit > interval of reals, in reasonable ways, preserving true analytical > results. > > In Parrradise you're naked and ignorant. If you perceive otherwise, is > it not Hell? > > There are NO known uses of transfinite cardinals in physics. > > Be a post-Cantorian, but it might be wise not to tell people that, > because they might not understand. > > Consider the ball and vase as a related rate problem. The vase is a > bucket with a hole in it, and it overflows. Find something in the > actual real world, not the fantasy "here's a finite vase that does not > cover the entire galaxy that can contain any finite number of balls", > which could never exist. Find an actual particle/wave system to which > to apply Zeno, and then maybe you'll learn about HUP, and how it > doesn't always apply, photons, time, and some of the notions about the > continuum of time. -- David Marcus
From: imaginatorium on 16 Oct 2006 15:42
Lester Zick wrote: > On 16 Oct 2006 08:07:56 -0700, imaginatorium(a)despammed.com wrote: > > > > >Han de Bruijn wrote: > >> Virgil wrote: > >> > >> > In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, > >> > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >> > > >> >> I find the idea absurd that natural numbers can be built > >> >>by putting curly braces around the empty set. > >> > > >> > It appears as if much of useful mathematics is ultimately based on what > >> > HdB finds absurd. > >> > >> The natural numbers can be defined without employing set theory. > > > >I should think they could be. Though I fancy the natural numbers will > >never be properly defined by anyone who is incapable of understanding > >the set theoretic definition. > > Well, Brian, if by "incapable of understanding . . ." you mean > "unwilling to agree with set theoretic assumptions regarding > definition of the naturals" I would certainly have to disagree. Well, I don't. By "incapable of understanding the set theoretic definition" I refer to anyone who lacks the capability for abstract thought required to understand the basics of set theory. I hesitate to say simply "anyone too stupid", though it is true it would be shorter. Brian Chandler http://imaginatorium.org |