An uncountable countable set
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From: Tony Orlow on 14 Sep 2006 12:06 Dik T. Winter wrote: > In article <4506dbd2(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > ... > > Yes, it should, and Wolfgang's and my position is that N is unbounded > > but finite, given that it only contains finite values, and only one per > > unit of value range. > > By the definition of "finite", "unbounded" and "finite" are in contradiction > with each other. So a number can not be both. Given the standard definition of an infinite set, and its "equivalence" with finite quantities, that's true. > > > Since the value range is finite, given that no two > > naturals are infinitely different, and given that only a finite number > > of elements can be fit sparsely in a finite range, the set is finite, > > even though it's unbounded and infinite by the Dedekind definition. > > Not the Dedekind definition, but the standard definition. There are > models with sets that are infinite, but Dedekind finite. (But in that > case you do not have AC, because AC implies the two kinds are the same. Do you mean finite but Dedekind infinite? That's what I'm proposing. It sounds like you're talking about some opposite approach. > > > At which bit position can the string achieve an infinite value? None > > that exists in that string. > > Indeed. And that is why you need to do something to give actual meaning > to such strings. In the 2-adics such is done, and in that case, > ...111 is sum{n = 0 -> oo} 2^n = lim{n -> oo} (1 - 2^n)/(1 - 2) = -1 > because in 2-adic metric lim{n -> oo} 2^n = 0. (The metric is defined > as d(a, b) = 1/2^n if the n-th digits of a and b are the highest order > digits where they differ.) That would appear to support my position that the number line CAN be viewed as a circle, that there is a generalization from 2's complement to n-base systems where numbers form a circle. Having looked up p-adic arithmetic, I gleaned that using prime numbers solves some problems. But, the T-riffics, solve some additional problems. :) > > > > This should be a rather big clue that something's wrong with your > > > concept, and that it does not, in fact, "work as a natural". > > It does indeed not "work as a natural", but it works as a 2-adic. If all p-adics are provably finite, then the distinction becomes transparent. > > > > Which presumably requires some extended Peano axioms in order > > > to exist? > > > > No, it just requires a loosening of the requirement that we only look at > > the minimal set satisfying those axioms. > > But it is useful to look at the minimal set satisfying those axioms. There > are too many sets that satisfy those axioms so that it is difficult to > state anything with sense when the sets are not minimal (unless they are > special). Howmany other sets can you formulate, which satisfy those axioms? > Take the Gaussian integers. They satisfy the axioms when we > state the successor function as S(k + i.l) = k + 1 + i.l Take the normal > integers, they also do satisfy the axioms with the standard successor > function (+1), take the rational numbers, they also do satisfy the axioms > when we define the successor of p/q as p/q + 2. Yes, it's very useful to classify number systems according to the operations which they allow and the relations between those operations. > > > I don't see that it contradicts > > them as they stand. > > It does not contradict them, but it is the minimality that makes things > provable that otherwise would not be provable. That depends on the acceptance of the limit ordinals as a model for the naturals, really, when it comes right down to it. To say the least ordinal for all finites is infinite is a mistake, in my opinion. > > > Extensions of the naturals such as this also fit > > into that model. > > Yes, like a host of other extensions. Well, then, if it doesn't contradict basic notions of the natural number, then it should be allowed as a model, and objections are non-logical, eh? Tony
From: Tony Orlow on 14 Sep 2006 12:17 Dik T. Winter wrote: > In article <4506d1ae(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > David R Tribble wrote: > ... > > > Start with a simple proof: Take the set of naturals, N = {0,1,2,3,...} > > > and remove one element to get set S = {1,2,3,...}. Now show that > > > the "T-size" of N is exactly one less than the T-size of S. In other > > > words, find a way to show that every counting of S versus every > > > counting of N always leaves one element of N (0) left over. > > > > Use IFR. N maps to S using f(n)=n+1. The inverse of that function is > > g(x)=x-1. So, over the range of 0 to N, |S|=|N|-1. Map N to the Evens E > > using f(n)=2n. The inverse function is g(x)=x/2, so over the range of N, > > the evens have |N|/2 elements. Isn't that intuitively satisfying? And > > gee, it works for finite sets accurately too!! > > How many elements has the set of primes? There is no well-known function that maps n to the nth prime, so IFR doesn't apply. Do you have an inverse function that specifies the nth prime for all neN? Didn't think so. However, the asymptotic density of the primes in [0,n] is n*ln(n) (not sure how this is proved), so I would say that the set has size N*ln(N). Or consider the following mapping: > for each n in N write n as prod p_i^e_i where the p_i are prime, now map > n = prod p_i^e_i to prod p_(i+1)^e_i. > what is the size of the resulting set, and why? (Oh, I should state that > 1 maps to 2. And, yes, there is a reverse mapping.) What is the function that gives the nth prime number? :( Tony
From: Tony Orlow on 14 Sep 2006 12:34 Dik T. Winter wrote: > In article <45070bcb(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > ... > > What about an unboundedly large but finite number? > > Let me ask you to, before I can answer such a question. What is your > definition of "number"? (I asked the same from Wolfgang Mueckenheim, > but his answer was not satisfactory, also not to himself, I think, > because he never answered to questions about it.) Let me start with a geometrical expression of quantity: distance. Certainly, if two points are different, there is some distance between them. Along a line, this distance is some number which is the distance multiplied by the number of units per distance. Now, that distance can be some discernible count of units, plus or minus 1, or it cannot. It it's less than any finite number of units, because it would require in infinite set of 0 bits to the right before getting to the first 1 bit, then it's infinitesimal. If it requires more than any finite number of bits to the left of the digital point, then it's infinite. This all depends on whether the most significant digit in the string is in a finite position, or in a positively or negatively infinite positions, in the string. > > Once you have answered that question we can probe further into the > definition of "unboundedly large but finite number". Okay. Did I? > > As far as I understand, for any kind of numbers, they are fixed, so > by definition not unbounded. Even if they have infinitely-indexed bit positions? Tony
From: Tony Orlow on 14 Sep 2006 12:38 Dik T. Winter wrote: > In article <1158102358.446630.158500(a)i3g2000cwc.googlegroups.com> "David R Tribble" <david(a)tribble.com> writes: > ... > > [http://en.wikipedia.org/wiki/Number] > > A number is an abstract entity that represents a count or > > measurement. In mathematics, the definition of a number > > has extended to include abstractions such as fractions, > > negative, irrational, transcendental, and complex numbers. > > > > I'd also add algebraic numbers, ordinals, cardinals, quaternions, > > octonions, matrices, tensors, p-adics, hyperreals, and > > surreals to that list. > > Somthing like that. In my opinion, when you have a set of objects, two > operations (+ and *) of which one is distributive over the other, > you can talk about numbers. But perhaps some more properties are > required? I have no idea. But that is only my opinion. In mathematics, > unless the context is clear, the word "numbers" is always qualified. > There is *no* mathematical definition of number. We can also include exponentiation and tetration, in which case we close or complete the circle of operations, in a sense. > > > You can see my confusion about Mueckenh's definition of "number". > > The only definition I have seen is where he stated that a "number" > should be larger, equal to or smaller than natural numbers (or > something like that). (That is, the trichotomy.) Alas, his > definition failed its purpose, to exclude the ordinals from the > class of numbers. Every raw quantity should be greater than, equal to, or less than, every other quantity, or in a 2-dimensional space, the same should hold for every distance. Tony
From: Tony Orlow on 14 Sep 2006 12:50
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > >> In article <1157743126.475610.189460(a)d34g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >> > Dik T. Winter schrieb: >> > > > > > A law is derived from the natural properties of arithmetics. >> > > > > >> > > > > Oh. What law derived from the natural properties of arithmetics is he >> > > > > talking about when he gets the completely determined set of all integral >> > > > > finite numbers? >> > > > >> > > > That is but his conviction. >> > > >> > > I ask you what law, but you are not willing to answer? Again, what law is >> > > he using (note that in English law generally refers to Theorem). >> > >> > Cantor held the opinion that there are some natural "laws" or "rules" , >> > not theorems, but Grundwahrheiten, so say truths, which are valid for >> > the natural numbers and which cannot be changed without leading to >> > rubbish. >> >> Yes, in English such things are called axioms. > > An axiom can be chosen arbitrarily, either the axiom or its negation. > Cantor's truths cannot be chosen but exist prior to any mathematics. > >> Although I disagree that >> changing them leads to rubbish. > > because you have not yet understood what Cantor's truths are. > >> Changing them leads to at most a >> different kind of mathematics. >> > Therefore, the truths are *not* axioms. Changing these truths leads to > rubbish. > >> > Something like axioms but not arbitrarily stated but derived >> > from "nature" or "reality". >> >> Ah, like the parallel postulate of Euclides. >> >> > One of them is 2 + 2 = 4 (in decimal >> > notation). Another one, according to his opinion, is the existence of >> > infinitely many finite numbers. >> >> Well, if you want to use only Cantor's view you should not attack current >> set theory. > > I have only tried to explain the difference between Cantor's view and > what you think what was Cantor's view. > >> In current set theory that is an axiom, and so I think my >> translation with the word "axiom" is quite correct. > > Then you should try to improve your thinking. You wanted to translate > Cantor and you have completely neglected his words and his view. A > worse translation is impossible. > > Regards, WM > Hi Wolfgang - I have renewed sense of faith in what you are trying to accomplish, given your faith to Cantor's original vision. I do think he grappled, in the face of fierce opposition, with fundamental concepts of infinity. I think he suffered for even attempting to answer such questions. Now that he has laid a groundwork, it's not unreasonable that it be improved. God Bless T |