Cantor Confusion
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From: Dave Seaman on 23 Jan 2007 19:02 On Tue, 23 Jan 2007 23:34:03 GMT, Andy Smith wrote: > David Marcus writes >>> But, to cut to the chase, if you don't have enough integers to define >>> even one real, what chance of counting them all? >> >>What in the world do you mean by "don't have enough integers to define >>even one real"? Do you have some problem defining sqrt(2)? >> > I meant that the infinite non-repeating irrational binary expansion of > sqrt(2) requires an actually infinite set of numbers to define its > location on the line (you don't buy my idea of conceptual numbers, so > just regard all reals as necessarily requiring an infinite number of > bits to describe them). > Since the integers are finite, you cannot represent a real requiring an > actually infinite number of bits, is what I meant. Maybe that is too > simplistic? Nonsense. There are infinitely many integers. To be precise, there are denumerably many integers. That's exactly as many as are needed to define one real, no more and no less. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: Virgil on 23 Jan 2007 19:07 In article <MPG.20202628da5b84d9989bf2(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > G. Frege wrote: > > On Tue, 23 Jan 2007 12:52:11 -0500, David Marcus > > <DavidMarcus(a)alumdotmit.edu> wrote: > > > > > There is some perverse pleasure in winning an argument with someone even > > > if the other person has no clue that you have won. And, the fact that > > > you are winning is much clearer (at least to you and to intelligent > > > bystanders) when arguing about math than when arguing about, say, > > > politics. > > > > Hmmm... hmmm... I think the point (for me) is that *I* am interested > > in the matters itself we are talking about (here). With other words, > > when discussing a topic I am aiming for (the) truth, not for winning > > an argument. (At least this is my "ideal".) > > A worthy ideal. > > > >> [...] Surely you will have noticed that there is (literally) NO progress > > >> when arguing with WM. (Imho it's extremely nonsensical to "argue" with > > >> such a guy.) > > >> > > > I basically agree, although whether it is extremely nonsensical depends > > > on the goal. What do you suggest we do with cranks? Just ignore them? > > > > Basically, yes (I'd say). > > > > Torkel Franzen once wrote (in this NG): > > > > "[...] Wolfgang M�ckenheim is a classic crank. Why do you imagine, as > > you seem to do, that there is any point arguing with him?" > > OK. If everyone else ignores him, I will too. However, if we (meaning > people who understand math) are going to carry on a public discussion > with him, I don't think it hurts if I toss in a few posts to help "our > side". Nothing worse than having a public argument with a crank and have > people who don't know much mathematics conclude that a real discussion > is going on. In other words, if we publicly argue with a crank, we > should win. Of course, "winning" does not require that one convince the crank himself of anything, it only requires convincing the vast majority of lurkers of the crank's crankhood. In which task the crank often unconsciously cooperates.
From: Andy Smith on 23 Jan 2007 19:15 David Marcus writes >> > >> >And if you want to make progress, have respect for technical terms. >> >"Uncountable" means something very specific. Does 1/7 have an "actually >> >infinite binary representation" in your terms? Decimal fractions for >> >"most" rational numbers go on without ending - but the rationals are >> >not uncountable. >> > >> Sure. But (almost) all real numbers have a random infinite bit sequence, >> and any one instance of those requires an infinite number of bits to >> specify it (is what I meant). > >Why should that observation have anything to do with whether the reals >are countable or uncountable? > It is just address space isn't it? You can't address 2^64 numbers with 32 bits, and you can't address 2^(actually infinite) numbers with a finite number of bits? OK, some numbers like the rationals and algebraic numbers have compact representations, but they are 0% of the total ... OK, that is hand waving, not a proof, but there is a clear distinction between finite and actually infinite ...? -- Andy Smith
From: Andy Smith on 23 Jan 2007 19:27 > >Since the integers are finite, you cannot represent a real requiring an >actually infinite number of bits, is what I meant. Maybe that is too >simplistic? > > To be more explicit, to represent (= address) all the reals in say [0,1] you would need as many bits for your integers as the reals occupy .. But that would require integers with an actually infinite bit length e.g. say, the reflection of the reals about the decimal point to give "numbers" like ...1101 Which is where I came in (with what's the problem with enumerating the reals?) but I am now better informed - . If all members of N are finite, there is no prospect of addressing the reals, so no surprise there.... -- Andy Smith
From: David Marcus on 23 Jan 2007 19:40
Andy Smith wrote: > > >Since the integers are finite, you cannot represent a real requiring an > >actually infinite number of bits, is what I meant. Maybe that is too > >simplistic? > > > To be more explicit, to represent (= address) all the reals in say > [0,1] you would need as many bits for your integers as the reals occupy > . But that would require integers with an actually infinite bit length > e.g. say, the reflection of the reals about the decimal point to give > "numbers" like ...1101 > > Which is where I came in (with what's the problem with enumerating the > reals?) but I am now better informed - . If all members of N are > finite, there is no prospect of addressing the reals, so no surprise > there.... Are you sure you are better informed? What you wrote is nonsense. Every real number has a binary expansion (and a decimal expansion). So, what in the world are you trying to say? -- David Marcus |