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From: David R Tribble on 31 Oct 2005 18:30 Albrech Storz wrote: > Do you have a test on objects to proof if they are sets or not? Or is a > set just that what you want to have to be a set? A set is a collection of entities taken as a whole. The entities are members of the set. Of all the possible entities, there are those that are in the set, and those that are not in the set. > What's your definition of sets? Is the water in a can a set of water? We can model the can as a set containing water molecules, but we must be able to distinguish each molecule from every other molecule, i.e., each molecule must have some unique numbering or comparison operator defined for it. We could, for example, assign each molecule a unique 3-dimensional coordinate within the can, but that won't work very well since the molecules move around and change their positions. It might work if the water were frozen. But a collection of water molecules is more like a "bag" than a set. A bag (multiset) is like a set but can contain multiple instances of the same element. Since all water molecules look alike, this would be a better model than a set.
From: David R Tribble on 31 Oct 2005 18:48 David R Tribble said: >> If that's the case, at what point do the naturals on the right side >> stop being finite and start being infinite? > Tony Orlow wrote: > Oh geeze. Here we go again. "No largest finite!!!" No kidding. There is no > single step where that happens, as you well know. Let me ask you this. > At what point does the count of naturals become infinite? At the point that all the naturals are enumerated (or, equivalently, are placed into a set)? Since there are an infinite number of them, the count of them is infinite. > That's the point at which the element values become infinite. (sigh). At what point is that? After the largest finite? Are those infinite element values also part of the set?
From: David R Tribble on 1 Nov 2005 13:18 David R Tribble said: >> We are forced to conclude that there is no natural s that maps to >> *N, and that therefore your mapping scheme is not a bijection >> between *N and P(*N). > Randy Poe said: >> I just want a member of *N to map to every >> element of P(*N), since you claim you have a map that maps >> some element of *N to map to every element of P(*N). > (For those of you scoring at home, *N is Tony's set of naturals which includes all the finite and infinite naturals. It's similar to the subset of positive integers from set *R of non-standard analysis, but obviously not quite the same.) Tony Orlow wrote: > If every natural maps to a set which does not contain it, then > you are asking for a natural which maps to the entire set, which requires an > end to the set in order to have a specific answer. Since the set doesn't end, > the natural mapped doesn;t either, and is .....1111. Why do you reject this > answer? I say this value is in *N. Prove me wrong. What you don't comprehend is that ...111 (which we'll call s) is the infinite sum of an infinite number of _finite_ terms: s = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + ... s = 1 + 2 + 4 + 8 + 16 + ... Therefore s has a 1 bit for each finite natural in N. Thus s maps to the entire infinite set of naturals N. Thus your mapping can be used to biject *N to P(N). However, s does not have enough bits to map to any of the infinite naturals in *N. So s cannot map to the entire set *N. Try it for yourself; list each bit of s (each power of 2 in s) and show the corresponding subset of P(*N) it maps to; you won't find any infinite naturals in any of those subsets. If s were to have enough bits to do this kind of mapping, it's clear that s would itself be a member of *N, and thus some bit or bits within s would map to s itself. This is clearly impossible because log(x) < x for all x (even infinite x, as per your TOmatics), so there does not exist an s large enough to map to itself, let alone to the entire set of infinite naturals in *N. >> 1. Do you have a bijection from *N to P(*N)? >> >> 2. Do you have a map that maps some element of *N to every >> element of P(*N)? > > Yes, every identifiable one. The entire set maps to ......11111. Remember, I > get to use infinite naturals in my *N. > And all of those infinite naturals are sums of finite powers of 2, and therefore cannot be used to map to any of the infinite naturals in *N themselves. You really don't see this, do you? >> 3. Are you aware you can't have #1 without #2? > > Sure. Are you aware that asking for the last element of an unending set > is a bogus demand? We're not asking for a mapping to the last element, we're asking for a mapping to the entire set *N, which is a member of P(*N). You alone seem to think this somehow requires a "last" member of *N.
From: David R Tribble on 1 Nov 2005 13:38 David R Tribble said: >> Let: >> n = 1 + 1 + 1 + 1 + ... >> and: >> s = 1 + 2 + 4 + 8 + ... >> >> Now let's group the partial sums of n: >> n = 1 + (1 + 1) + (1 + 1 + 1 + 1) + (...) + ... >> n = 1 + 2 + 4 + 8 + ... >> >> Obviously, we can group each set of 2^p 1's in our infinite sum so >> they are identical to the terms in the series for s. So n = s. > Tony Orlow wrote: > Your rearrangement violates the rules for infinite series. You don't understand infinite sums very well. It works going the other way, too: n = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + ... and: s = 1 + 2 + 4 + 8 + ... s = 1 + (1+1) + (1+1+1+1) + (1+1+1+1+1+1+1+1) + ... s = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + ... so: s = n Here are some more to play with: n = 1 + 1 + 1 + 1 + ... n-1 = 0 + 1 + 1 + 1 + ... n+0 = 0 + 1 + 1 + 1 + ... so: n-1 = n+0 n-1 = n And: n = 1 + 1 + 1 + 1 + ... n+2 = 2 + 1 + 1 + 1 + ... n+2 = (1+1) + 1 + 1 + ... n+2 = 1 + 1 + 1 + 1 + ... so: n+2 = n And so on, for all these divergent sums (which all happen to be equal). Isn't this fun? Tony Orlow wrote: > No, the sum of an infinite number of 1 is not all 1's, but the unit infinity, > 1:000...000. Do you somehow think the sum of any number of 1's in binary is a > string of 1's? It's not. Then what is the digital form of N = 1 + 1 + 1 + 1 + ... ? Is it less than S = 1 + 2 + 4 + 8 + ... ? What is your 1:000...000 - 1? Isn't it supposed to be your N-1? Or is it actually your 2^N-1?
From: David R Tribble on 1 Nov 2005 15:20
David R Tribble said: >> You've offered plenty of statements about "unidentifiable largest >> naturals" and "set ranges" and "unit infinities", but you've never >> actually pinpointed how you get from the finites to the infinites. > Tony Orlow wrote: > Here's a nice long one, with plenty to chew on: > > Among the Axioms of Finiteness (which probably exist already, somewhere): > > For A and B finite (non-zero) and infinite, A^B or A*B is either infinite or > finite according to the following table- > > A Finite Infinite > ----------------------------------- > B | > Finite | Finite Infinite > Infinite | Infinite Infinite > > Note that this table is equivalent to the OR truth table, when you replace > "finite" with 0 and "infinite" with 1. In other words, the result is infinite > if either A OR B is infinite, and is finite if A AND B are finite. Great. So combining an infinite value with any other value always results in an infinite value. Now you need to define "infinite value". > With a set of symbols of size S, one can create a set of unique strings of > length L, whose maximum size is N=S^L. > > When we combine these two "axioms", we see that N=S^L is only infinite for > either infinite S or L. Therefore, reapplying what our symbols mean to > "reality", we can only have an infinite set of strings if we either have an > infinite set of symbols with which to build them, or allow infinite length > strings. So an "infinite value" is a string of digits that has an "infinite length". Now you need to define "infinite length". > Now we introduce the digital number systems, defined as usual, with S equal > to the radix, a finite whole number greater than 0, the symbols in the set > representing the whole numbers from 0 to the radix minus 1, and each string > position representing a power of the radix that increases going leftward, > with a digital point directly to the right of the digit place representing > the zero power of the radix. The value represented by the digital number, > V, is given by V=sum(i=-oo->+oo: a_i * b^i), where a_i is the digit at digit > place i, and b is the radix. That sounds about right, doesn't it? Did I miss > anything? Okay, that's the radix representation of numbers. > Now, if i is infinite and positive for a given digit, then it is infinitely > far to the left of the point, and represents the radix to an infinite power, > right? > So, given our axiom of finiteness, we can say that a finite radix to an > infinite power gives an infinite result, right? I don't know. What do "is infinite" and "is infinitely far to the left" mean? Is this like your "infinite length" that you never got around to defining? For comparison, Dedekind started by first defining "infinite set", and then defining "finite set" as "not infinite set". You seem to be going the other way, defining "finite number" first, then somehow allowing for "infinite number". Somehow. > [snip] > So, we have three axioms (actually it needs to be about 10, right, to cover > the digital numbers ?): > > 1. N=S^L > > 2. A^B and A*B are only finite for finite A and B, and otherwise infinite. > > 3. A digital number with nonzero digits in digit places infinitely to the > left of the point represents an infinite value. You are simply assuming the existence of "infinite" values. You have not shown how they are defined (brought into existence), except by refering to "infinite" lengths, which you have also not defined. So it looks like you are simply assuming that infinite values exist, and that finite values form some special subset of them. > So, > > infinite(N)->infinite(S) OR infinite(L) by 1 and 2 > NOT(infinite(S))->infinite(L) by definition of radix > infinite(L)->infinite(V) by formula for V > > Hopefully this is step-by-step enough to get SOMEBODY to understand this line > of reasoning. We understand. The problem is that it's not complete. How do your rules allow infinite numbers to be created? How do you get from the finite numbers to the infinite numbers? If I have a number, how do I tell if it has an infinite length? If I have two infinite values, how do I know which one is larger? Do your infinite sums result in different values? If so, you ought to show how this can be, since it contradicts normal arithmetic. |