An uncountable countable set
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From: Tony Orlow on 25 Sep 2006 20:37 Han de Bruijn wrote: > Tony Orlow wrote: > >> Han de Bruijn wrote: >> >>> Tony Orlow wrote: >>> >>>> Han, if I prove inductively, say, that 2^x>2*x for all x>2, do you >>>> find it objectionable to say that this also applies to any infinite >>>> value, if such a thing existed, given that any infinite value would >>>> be greater than any finite value, and therefore greater than 2? >>> >>> Give me one reason, Tony, why I would find such a theorem interesting >>> in the first place. I'd prefer the ultimate terseness in mathematics, >>> especially if it comes to infinities. >> >> Okay, but it's not that particular theorem that is of interest, but >> the system of theorems this extension of induction makes possible. If >> we can say that such inductive arguments hold in the infinite case, >> and within any range up to n one set has size 2n (multiples of 1/2) >> and the other x^2 (logs base 2 of naturals), then proving that n>2 -> >> n^2>2n proves the second to be bigger than the first, for infinite n, >> since they are larger than 2. This gives us a nice exact way of >> comparing infinite sets over an infinite range, free from "limit >> ordinals" and other nonsense. This allows us to distinguish between >> the sizes of the naturals vs. the evens, and even detect the addition >> or removal of a single element from an infinite set. >> >> Hope that made sense. > > Thought I've made my point. When I said that your infinities are not > worse than their standard alephs, then I didn't say that I find your > infinities "better" :-( Sorry ... > > Han de Bruijn > I guess that's a no. :| Tony
From: Tony Orlow on 25 Sep 2006 20:46 Han de Bruijn wrote: > Tony Orlow wrote: > >> Mike Kelly wrote: >> >>> Han de Bruijn wrote: >>> >>>> Mike Kelly wrote: >>>> >>>>> Han de Bruijn wrote: >>>>> >>>>>> Mike Kelly wrote: >>>>>> >>>>>>> What the hell are you talking about? Arguing with someone who can't >>>>>>> speak English is getting aggravating. >>>>>> >>>>>> My English is much better than your Dutch. >>>>> >>>>> So what? Your English is still too poor for this discussion to be >>>>> fruitful. >>>> >>>> Still don't get the point, huh? >>>> >>>> You are lacking even the most elementary form of politeness. It's very >>>> impolite to cut of a discussion with somebody from a foureign country - >>>> somebody who is doing his best to communicate with you - only because >>>> you are obviously superior in expressing your thoughts within your own >>>> mother's tongue. >>> >>> You're a very rude person yourself, Han. I generally don't feel the >>> need to be civil to those who won't reciprocate. >> >> I don't think I have ever found Han to be rude, except when he >> referred to my "babbling" recently. Ahem. But anyway, while we >> disagree on the actuality of any infinity, we have the open mind of >> spirited debate, and feel no need to get nasty. > > I may be rude sometimes, but I never get _personal_ by calling somebody > an "idiot" or a "crank". Tony's "babbling" translates with Euroglot as > "babbelen" in Dutch, which is a word I can use here in the conversation > with my collegues without making them very angry (if I say "volgens mij > babbel je maar wat"). But, of course, I cannot judge the precise impact > of the word in English. Apologies if it is heavier than I thought. Do you have the saying, "Shallow brooks babble, and still waters run deep"? I figured you picked up the usage from this forum, actually. It's meant, in English, to mean you aren't making any sense. :) > >> Furthermore, I have never had any trouble understanding what Han is >> saying, except where he is using some mathematical construct with >> which I am not familiar. His English is not bad, and blaming your >> disagreement on his inability to communicate is kind of low. > > Thank you very much, Tony, for this sort of defense. My pleasure. It seemed like a vacuous excuse. I get pretty sick of those diversionary tactics. > >> So, let's engage in lively debate, and maintain our civility, while >> chopping each other's arguments to pieces. Of course, this can only >> happen if we don't consider our arguments to be part of our anatomy. >> Otherwise, it gets personal. >>> >>>>> You are misinterpreting virtually all my posts. You claim that you're >>>>> not dishonest so I have to conclude you're simply incapable of >>>>> comprehending written English. This makes this whole subthread >>>>> pointless. >>>> >>>> I have only this kind of trouble with _you_ and nobody else on the web. >>> >>> Really? You've never had anybody else other than me complain that you >>> misinterpret their posts? I suppose I must have hallucinated dozens of >>> posts I've seen of just that, then. >>> >>> You've never had anyone other than me struggling to understand what the >>> devil you mean by your broken English? I must have hallucinated, for >>> example, "A little physics would be no idleness in mathematics", then >>> :)? >> >> Well, that's a difficult type of quote. Han - I wouldn't mind working >> on exactly how you want to say that in English, if you like. :) > > Uhm, since litteraly everybody is complaining ... Let it be an encrypted > message then :-) Well, it seems to me that perhaps you're saying something like, "Those with their heads in the abstract should keep their feet in the concrete", though that sounds a little funny. Math=Science? > > Han de Bruijn > Tony
From: Tony Orlow on 25 Sep 2006 20:51 Han de Bruijn wrote: > Mike Kelly wrote [ snipping all of the awful nesting ]: > >> Tony Orlow wrote [ .. snip .. ]: >> >>> which led him to say, "Precisely!". Hmmm, I don't think it's an English >>> problem. >> >> Well apparently neither of you have any idea what my point was. >> Hopefully I have clarified it now. > > That's funny. So the "English problem" is now between two native English > speakers and a Dutchman. Who is misunderstanding who ... > > Han de Bruijn > Hahaha! Whew! That was good. :) Thanks, Han Tony
From: Tony Orlow on 25 Sep 2006 21:06 David R Tribble wrote: > [I'm a bit late in this subthread, but no matter...] [ I just got back to it today too] > > Tony Orlow wrote: >> If omega is the successor to the set of all finite naturals, ... > > It's not. It's a limit ordinal. > Yes, it has no predecessor, but it is what comes immediately after the complete set of finite naturals, which has no last. I don't believe they make any sense anyway. :) > >> ... it is >> greater than all finite naturals, as any successor is greater then all >> those that precede it. It is certainly a positive number, if it is a >> count or size of anything. If it is a positive natural greater then 1, ... > > No, it can't be a natural, because of what you just said, that's it > larger than all naturals. A natural can't be larger than itself, so > omega can't be a natural. > I know the problem here is that it is not successor to anything, so that it can be a smallest infinite. But, my point is that it's "greater than any finite natural" in the sense that, in the well-ordered set of ordinals, it comes after all the finite successor ordinals. Why isn't it true that, if x comes before y in the ordinals, x<y is not necessarily true? I see no reason why this doesn't imply omega is "larger than any finite value". > >> ... then its reciprocal is a real in (0,1). > > Nope. If w > x for all real x, then 1/w < 1/x for all real x as well. > Which means that 1/w cannot be a real. Not a standard real. I'm talking here about nonstandard infinite real values. Obviously those aren't standard reals. Any inverse would be infinitesimal, and therefore also not a standard real. So? > > If 1/w was a real, then 1/(1/w) = w would also be a real, > because 1/x is real for all real x > 0. But then w could not be > greater than all reals (or all naturals), since it would have to be > a real that is greater than itself. > For standard reals, that's all very correct. > >> To say that some count which is >> greater than any finite count does not obey this general rule is a >> kludge, like all the transfinite "arithmetic". > > Your "count" is not a well-defined term. Do you mean real, ordinal, > cardinal, or something else? > For the sake of this argument, we can talk about infinite reals, of which infinite whole numbers are a subset. Tony
From: Tony Orlow on 25 Sep 2006 21:17
mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > >>>>> So lim [n-->oo] n/aleph_0 < 1 >>>> Division is not defined for infinite cardinal numbers. >>> Is that your only escape? If you dare to say that aleph_0 > n for any >>> n e N, then we can conclude the above inequality. >> No, because division is not defined on infinite cardinal numbers. The >> above inequality is meaningless. > > For cardinals we have well defined n*aleph_0 = aleph_0 for any natural > n (see any book on set theory). Multiply with 1/n for any natural n, > then you get well defined > > A n e N: aleph_0 / n = aleph_0 > 1. Hence, my followin statement is > correct. >>> But remedy is easy. >>> Take lim [n-->oo] aleph_0 / n > 1 and reverse the following fractions >>> analogously. > [...] > >> Your position seems very inconsistent. You claim that numbers have no >> existence outside their representation. And now you are claiming there >> exists a "true" arithmetic. > > It is obvious. You can verify it by experiment: II + III = IIIII. > > [...] >> It is not a proof. Division is not defined where either operand is an >> infinite cardinal number. > > But you can conclude n / aleph_0 < 1 by inserting aleph_0 > n which is > definied *if aleph_0 is a number in trichotomy with natural numbers*. > > You cannot have both, assert that aleph_0 is a number larger than any n > but on the other hand prohibit that the inequality n < aleph_0 be > utilized. > > Regards, WM > I agree. If x is infinite, and that means greater than any finite, and trichotomy holds, then 1/x is in [0,1], and is real, though infinitesimal, of course. But, I am getting the feeling that set theorists now don't want to claim that infinite is greater than finite. Hmmm... Tony |