An uncountable countable set
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From: MoeBlee on 19 Oct 2006 20:28 You wrote too many confused and uninformed things for me to even care to sort through. I'll take you up on your last line, though: > You > say you set theory texts define "cardinality" in a certain way which > is pretty much circular if relying on cardinality for equinumerosity. I don't say that. And the definition of 'cardinality of' does use 'equinumerosity', but the definition of 'equinumerosity' does not use 'cardinality of', so there is not the circluarity you just arbitarily claim there to be. I've already been over the subject of mathematical definitions with you in other threads. But please do consider all your points, objections, and conceptions to be vindicated by my increasing apathy to try to bring you to reason about anything at all. MoeBlee
From: Ross A. Finlayson on 19 Oct 2006 20:37 Lester Zick wrote: .... > > So what is it exactly that "set" theory allows us to do in mathematics > that we couldn't already do without it? Define infinity? Define > regularity? Define choice? Define ordinals? Define cardinals? You seem > to be of a psychological frame of reference prevalent among modern > mathematikers that arithmetic in the form of set theory represents > some kind of TOE. > > ~v~~ Lester, Only the null axiom theory could be the TOE. Ross
From: Ross A. Finlayson on 19 Oct 2006 20:42 David R Tribble wrote: > > Irrelevant. All of the reals can be written in digital form. But the > reals with non-terminating non-repeating fractions form the uncountable > set of irrationals that comprise most of the reals. Nobody ever has nor ever will write down a transcendental or non-rational algebraic real expansion digit by digit. A wide variety of those values can be written concisely, eg e, pi, gamma, etcetera. There are spigot and digit extraction algorithms where you can compute any digit of those numbers, that computation has finite cost as does storage of its return value. Ross
From: Ross A. Finlayson on 19 Oct 2006 20:46 MoeBlee wrote: > Han de Bruijn wrote: > > The confusion stems from the fact that I cannot and shall not understand > > the _infinite_ counterparts of the finite cardinals and ordinals. > > How can you understand if you won't read a book that explains it? (By > the way, Halmos is a good book, but it's just an overview; it doesn't > give you the full explanations that you need.) > > So you seem to think it is better to spout nonsense on the Internet > about a subject you cannot possibly understand since you insist that > you won't. > > MoeBlee Did you find that statement in the book about s? Skolemize, any model is countable. There is no "everything" in ZF. Ross
From: Ross A. Finlayson on 20 Oct 2006 02:35
David Marcus wrote: > Ross A. Finlayson wrote: > > David Marcus wrote: > > > Ross A. Finlayson wrote: > > > > Also, for each later time t less than noon, there are more balls in the > > > > vase than at time t, because for each difference there are ten balls > > > > added for each removed. > > > > > > > > So, how does the ball remove itself? > > > > > > > > Consider the ball and vase as a related rates tank problem. You pour > > > > in a dram of this syrupy liquid, and it goes to the bottom of the tank, > > > > where there's a hole in the bucket, Elvira, and the liquid drains out. > > > > That would seem similar to your problem here with the balls and vase, > > > > no? It even offers a mechanism, where you need to expand on some of > > > > the explanation there of the mechanics of the balls and vase. > > > > > > > > Your non-explanation is not a constraint, it's handwaving. > > > > > > > > So, the tank obviously fills. > > > > > > How would you translate the following problem into English ("balls", > > > "vase", "time")? > > > > > > Problem: For n = 1,2,..., define > > > > > > A_n = 12 - 1 / 2^(floor((n-1)/10)), > > > R_n = 12 - 1 / 2^(n-1). > > > > > > For n = 1,2,..., define a function B_n by > > > > > > B_n(t) = 1 if A_n < t < R_n, > > > 0 if t < A_n or t > R_n, > > > undefined if t = A_n or t = R_n. > > > > > > Let V(t) = sum{n=1}^infty B_n(t). What is V(12)? > > > > > > For n > 1, A_n < R_n < 12 for all n, so B_n(12) = 0 for all n, so, the > > sum of the B_n's is zero. > > Thank you. But, I didn't ask for your solution to the problem. I asked > how you would translate the problem into English, i.e., using the words > "balls", "vase", and "time'. So, please translate the problem into > English. > > > As Cauchy sequences: R_n is 12, A_n is 12, and B_n(12) is 0. > > > > Compute the difference of R_n - A_n > > D_n = 1 / 2^(floor((n-1)/10)) - 1 / 2^(n-1) > > > > = ( 0, 1/2, 3/4, 7/8, 15/16, 31/32, 63/64, 127/128, 255/256, > > 255/512, 511/1024, 1023/2048, ... ) > > > > Notice it's not monotonic, in that not D_n+1 < D_n. So, the standard > > limit doesn't exist, where people used to say monotonicity was a > > requirement. I'd agree the limit exists and is zero, because 12-A_n > > goes to zero and 12-R_n < 12-A_n. > > > > The required vase is infinitely huge. In fact, it would have to be so > > huge, the balls are already in the vase, else there's not room for > > them. > > > > In the parallel to the related rates tank problem, almost all the syrup > > overflows and is lost, besides that as an incompressible fluid the > > pressure would shatter any finite tank. > > > > A rabbit that runs unimpeded at 10 m/s is bound to a 1 m/s turtle, with > > the turtle trodding the other way the rabbit does 9 m/s. If your vase > > is empty you have that the turtle always beats Achilles if it starts a > > fraction ahead. > > > > Please explain the mechanics of the vase. How do you retrieve the > > selected ball at time t? > > > > Find a real world situation, with photons and a black hole or > > something, where you might actually explain the mechanics of this > > completion of infinity and etc., else it's quite unrealistic. > > I agree it is not realistic. > > -- > David Marcus Hi David, Well, I don't care to. That's apathy not distaste. You introduced that system, what's your point? Non-locality implies information transfer at greater than c, as you know, or, as they say, simultaneity. Faster than light, you know, superluminal, information transfer at greater than c. You see, in the theory of relativity, nothing goes faster than the speed of light. Ross |