An uncountable countable set
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From: Mike Kelly on 7 Sep 2006 10:16 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Dik T. Winter wrote: > >>> In article <44fdcaf1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > >>> > Dik T. Winter wrote: > >>> > > In article <1157367096.604428.36330(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > >>> ... > >>> > > > A number has only one of the following properties: It is larger than or > >>> > > > smaller than or equal to any natural number. > >>> > > > >>> > > So omega and aleph-0 are numbers. They satisfy the definition. Thanks. > >>> > > >>> > So, they are numbers which are larger than any finite number? Why then > >>> > do we not consider an inductive proof of the form E y e N A x>y P(x) not > >>> > to prove P(aleph_0) or P(omega)? > >>> > >>> That would be a new axiom, and you may consider it, but it leads to > >>> contradictions when you retain the current definitions of aleph_0 and > >>> omega. > >> Yes, I am well aware of that. That's why I have chosen to reject those > >> concepts in favor of finding something better, based on this > >> infinite-case induction. Noting personal. > >> > >> :) > > > > Better in what sense? What mathematics are you hoping to be able to do > > with your new foundation that cannot be done with ZFC? > > > > It accounts for changes of a single element between infinite sets and > therefore provides for a full spectrum of ordered infinite sets, thus > making the Continuum Hypothesis null and void. It relates the continuum > to the hypernaturals formulaically, and provides for an integration of > sets and measure in the infinite case. It rids us of anomalies like > omega-1=omega. > > If you remove an element, the proper subset should ALWAYS be smaller by > 1. That is the case for me. For a theory to claim a proper subset is the > same "size" as the proper superset is an immediate deal-breaker for me. So, the answer to my question is "It is better in that it satisfies Tony Orlow's intuitions". Have fun with that. Don't expect anybody else to give a toss. -- mike.
From: stephen on 7 Sep 2006 10:16 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: > given that all bit positions are finite, and that >> therefore no place in that string can achieve an infinite value, and >> that any such number has predecessor and successor. The cute thing is >> that the successor to ...1111 is 0, and that ...1111 is essentially -1. :) > Hmm. So -1 is "essentially" a lot larger than 1, for example, whereas > add 5 to both sides and you get the other thing. Well, that's faintly > amusing ice pose. Given that Tony apparently thinks that if you keep adding 1's, you eventually get back to zero, I cannot understand why he was so upset by the balls and vase problem. Stephen
From: Mike Kelly on 7 Sep 2006 10:20 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> And yet, you are saying it's not technically correct. So, 1 is a natural > >>>> but not a rational or real? If rigorous formulations come to that > >>>> conclusion, then rigor does not ensure correctness. > >>> You're hopeless. You didn't understand a thing I said, which may be my > >>> fault for not providing adequate explanation in the context of your > >>> ignorance of the subject; but it is no my fault that you won't even > >>> look at a book on mathematics, such as even a introductory text in real > >>> analysis. > >>> > >>> MoeBlee > >>> > >> Your whole point here is ludicrous. You are arguing that the naturals > >> are not a subset of the rationals, which are not a subset of the reals. > >> While each superset may require a more complex construction than the > >> subset, all elements of the subset are covered by the superconstruction. > >> As points on the real line, they are all real numbers. What you're > >> saying amounts to what I said above, which is indeed absurd. So, no, I > >> don't understand a thing you're saying, when you say naturals aren't > >> reals. Sorry. > > > > All natural numbers are finite ordinals. All real numbers are infinite > > sets of rationals. Obviously a set which contains only finite ordinals > > is not a subset of a set that contains only infinite sets of rationals. > > > > Both denote a point on the real line. Those set constructions are a > means to define the point. Different means of specification may define > the same point. "The number line" is a visual aid to help schoolchildren. What relevence does that have to a set of finite ordinals not being a subset of a set of sets of rationals? -- mike.
From: stephen on 7 Sep 2006 10:38 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >>> Your whole point here is ludicrous. You are arguing that the naturals >>> are not a subset of the rationals, which are not a subset of the reals. >>> While each superset may require a more complex construction than the >>> subset, all elements of the subset are covered by the superconstruction. >>> As points on the real line, they are all real numbers. What you're >>> saying amounts to what I said above, which is indeed absurd. So, no, I >>> don't understand a thing you're saying, when you say naturals aren't >>> reals. Sorry. >> >>> Tony >> >> int main() >> { >> int x=1; >> float y=1; >> } >> >> Does x equal y? >> >> Stephen > Yes, they even have the same bit string representing them (ignoring > length). Try bigger numbers next time. > Tony Really? The computer does not think they have the same bit string representing them. Why do you? int main() { int x=1; float y=1; printf("%d %d %08x %08x\n",sizeof(x),sizeof(y),x,y); } Stephen
From: Mike Kelly on 7 Sep 2006 10:50
stephen(a)nomail.com wrote: > imaginatorium(a)despammed.com wrote: > > > Tony Orlow wrote: > > given that all bit positions are finite, and that > >> therefore no place in that string can achieve an infinite value, and > >> that any such number has predecessor and successor. The cute thing is > >> that the successor to ...1111 is 0, and that ...1111 is essentially -1. :) > > > Hmm. So -1 is "essentially" a lot larger than 1, for example, whereas > > add 5 to both sides and you get the other thing. Well, that's faintly > > amusing ice pose. > > Given that Tony apparently thinks that if you keep adding 1's, you > eventually get back to zero, I cannot understand why he was > so upset by the balls and vase problem. > > Stephen I was actually thinking of that problem in the car a week ago. Surely even if there are "infinite integers" to go on the balls, those balls get removed infinitesimally before Noon? The nth ball gets removed at the nth iteration, at time -(1/2) ^ n. Surely Tony would argue that this is valid in the infinite case. Oh well. -- mike. |