An uncountable countable set
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From: David Marcus on 31 Oct 2006 23:50 Ross A. Finlayson wrote: > David Marcus wrote: > > Mike Kelly wrote: > > > Tony Orlow wrote: > > > > Where are the iterations mentioned there? You're missing the crucial > > > > part of the experiment. By your logic, you could put them in in any > > > > order and remove them in any order, and when you say both processes are > > > > done, nothing's left, but that's BS. It ignores the sequence specified. > > > > This is just a distraction. > > > > > > Yes, if you insert and remove exactly the same balls then you get the > > > same result when you're done, no matter what order you did it all in. > > > Why is that BS? It seems blindingly obvious. > > > > > > But I forgot, you think that if you shift all the insertions 1 minute > > > further back in time, you DO get an empty vase at noon, right? I really > > > don't understand how your mind works. > > > > Try the mental picture with the water. We fill it up, then we start > > letting it run out. No reason all the water shouldn't empty out of the > > vase by noon. > > > > -- > > David Marcus > > Hi, > > There are only sets in ZFC. When people talk about the real numbers in > ZFC, it is as a construction of sets in ZFC has been found to be > isomorphic in a relatively strong sense to the real numbers. > Obviously, mathematicians want to find the best representation of > everything the real numbers are or must be by their nature. > > Then, that gets into that some feel that mathematics can't assume what > it sets out to prove. That's reasonable. By the same token, the real > numbers have many, many roles to fill, and some of them have, for > example, in the projectively extended real numbers, points at > plus/minus infinity. > > It is relatively standard to define a real number as a Dedekind cut or > Cauchy sequence, which are basically defined in terms of sequences of > rationals, which resolve to generally the familiar decimal > representation which is adequate in finitely expressing rationals, and > with radicals, algebraics. > > If a Dedekind cut is as was recently stated some "initial segment" of > the rationals, I wonder to what ordering that pertains. > > Consider how that is to describe an irrational number. Basically the > sequence of elements is to converge towards the number. There's an > irrational less than one and greater than .9, in decimal, less than one > and greater than .99, less than one and greater than .999, etcetera. > The general consensus here is that .999... = 1, yet for each > .999...999, there is an irrational between it and one. So, does that > not seem that there are irrationals unrepresentable via > Dedekind/Cauchy? It would seem that certainly as the irrational is > some finite distance from 1 that it would be between .999...998 and > .999...999, and between that irrational and one are infinitely many > more numbers forever, there always exist irrationals between .999...999 > and 1, and, for Dedekind/Cauchy to represent them, they must have a > unique representation. > > For no finite number of 9's or rep-units in binary can these > irrationals in the diminishing remaining interval be represented, and > for any infinite number of rep-units the result is said to be one. So, > either between the finite and infinite those values are represented, or > they're not, and due to the completeness of the reals, if they're not, > then Dedekind/Cauchy, the standard set-theoretic method to construct > real numbers, is insufficient to construct some real numbers. > > For any it's so, for all it's not, or vice versa. Don't worry I've > heard of the transfer principle. > > Consider the representation of rational numbers, for example 9/10. > That would be .9, .90, .900, ..., .9(0): 9/10's. There is no last > element of that list, .9 could be an initial segment of a sequence for > 9/10's or any irrational between .9 and 1.0, as above. The initial > sequence .9, .90 could be an initial sequence for any irrational > between .900 and .91. The initial segment .900 could be an initial > sequence for any number between .9000 and .901. > > .90000 <= x <= .9001 > .900000 <= x <= .90001 > .9000000 <= x <= .900001 > .90000000 <= x <= .9000001 > .900000000 <= x <= .90000001 > .9000000000 <= x <= .900000001 > .90000000000 <= x <= .9000000001 > .900000000000 <= x <= .90000000001 > .9000000000000 <= x <= .900000000001 > .90000000000000 <= x <= .9000000000001 > .900000000000000 <= x <= .90000000000001 > ... > > As the number of zeros diverges, the diminishing interval goes to zero, > where the lower and upper bounds are a and b, lim n->oo b-a = 0. For > any finite iteration there are obviously a continuum of elements that x > could be, so for a value, x, to not obviously be among a continuum of > possible values there must be infinitely many iterations. > > Keep in mind that there are printed counterexamples to standard real > analysis with a least positive real. > > Obviously the ground around .999... vis-a-vis 1 is very well turned, > that's the point, to some extent we're talking about significant > ephemera. > > Look at the 1 on the right side above. Where does it go? > > .90001 <= x <= .9002 > .900001 <= x <= .90002 > .9000001 <= x <= .900002 > .90000001 <= x <= .9000002 > > Standardly, equal. > > .90001 <= x <= .901 > .900001 <= x <= .9001 > .9000001 <= x <= .90001 > .90000001 <= x <= .900001 > > .90001 <= x <= .91 > .900001 <= x <= .901 > .9000001 <= x <= .9001 > .90000001 <= x <= .90001 > > .90001 <= x <= .91 > .900001 <= x <= .901 > .9000001 <= x <= .9001 > .90000001 <= x <= .90001 > > .90001 <= x <= .91 > .900001 <= x <= .91 > .9000001 <= x <= .901 > .90000001 <= x <= .9001 > > .90001 <= x <= .91 > .900001 <= x <= .91 > .9000001 <= x <= .91 > .90000001 <= x <= .901 > > .90001 <= x <= .91 > .900001 <= x <= .91 > .9000001 <= x <= .91 > .90000001 <= x <= .91 > > On the left and right side each converges to 9/10, but as this > continues the lhs is .90 and the rhs is .91. > > So, is the vase empty at noon? Yep. -- David Marcus
From: David Marcus on 31 Oct 2006 23:54 Tony Orlow wrote: > stephen(a)nomail.com wrote: > > David R Tribble <david(a)tribble.com> wrote: > >> [Apologies if this duplicates previous responses] > > > >> Tony Orlow wrote: > >>> I am beginning to realize just how much trouble the axiom of > >>> extensionality is causing here. That is what you're using, here, no? The > >>> sets are "equal" because they contain the same elements. > > > >> Yes, the basic definition of set equality, the '=' set operator. > > > >>> That gives no > >>> measure of how the sets compare at any given point in their production. > > > >> This makes no sense. Sets are not "produced" or "generated". > >> Sets simply exist. > > > >>> Sets as sets are considered static and complete. > > > >> Correct. > > > >>> However, when talking > >>> about processes of adding and removing elements, the sets are not > >>> static, but changing with each event. > > > >> Incorrect. If we define set A as containing the elements a, b, and c, > >> then A = {a, b, c}. Period. If we then talk about adding elements d > >> and e to set A, we're not actually changing set A, but describing > >> another set, call it A2, that is the union of A and {d, e}, so > >> A2 = {a, b, c, d, e}. > > > >> Nothing is ever "added to" a set. Rather, we apply operations (union, > >> intersection, etc.) to existing sets to create new sets. We don't > >> change existing sets. > > > > Just like when we add 5 to 2 to get 7, we do not change the 5 or 2 > > to create a 7. Or when you celebrate a birthday, your age changes, > > but the number that represented your age does not change. A different > > number is used to represent your age, but the "old" number remains > > as it always ways. > > > > This idea of "changing" sets seems to be at the heart of a lot > > of people's misconceptions about set theory. > > > > Stephen > > Then it's an entirely different vase, a countably infinite number of > times over and over before noon, and if there is a vase at noon at all, > it's a different vase, with different balls. That seems to fit in with > your concept of set theory, eh? Ugh. Stephen said "set" and you reply "vase". They are different words. As such, what you wrote appears to be a non sequitor, unless you can connect it more directly. -- David Marcus
From: imaginatorium on 1 Nov 2006 00:06 Lester Zick wrote: > On Tue, 31 Oct 2006 10:30:08 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > > [. . .] > > Tony, I'm going to post several replies to this one post because I've > come up with a couple of ideas which may (or may not) appeal to you. > > First off why not change your approach in the following way. It seems > to me that you could arrange all the naturals on the x axis. Then > instead of trying to cram in all the transcendentals on the same axis, > try putting transcendental infinites on the ordinal y axis instead. > > However if you try this approach you may find that you need another > mutually orthogonal z axis to accommodate another class of infinites. > I don't know if this is going to work completely or not. But I think > it holds considerably more promise than trying to accommodate it all > on one more or less circular x axis alone. > > In any event this is the end of this particular suggestion. I hope it > helps and sheds some light on what I think is going on in mechanical > terms. In any event I'll get back to your original message now plus > what I think will turn out to be definitive mechanical arguments on > the subject of transcendentals and conventional linear analysis of the > reals. > > ~v~~ Oh Lester - you really are a hoot!! Out of curiosity, suppose the natural 2 is at (2,0) in conventional x-y coordinates, and pi (which I believe you agree is transcendental) is at say (0,7), whereabouts would 2pi be? Brian Chandler http://imaginatorium.org Keep the poetry flowing...
From: stephen on 1 Nov 2006 12:38 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> David R Tribble <david(a)tribble.com> wrote: >>> [Apologies if this duplicates previous responses] >> >>> Tony Orlow wrote: >>>> I am beginning to realize just how much trouble the axiom of >>>> extensionality is causing here. That is what you're using, here, no? The >>>> sets are "equal" because they contain the same elements. >> >>> Yes, the basic definition of set equality, the '=' set operator. >> >>>> That gives no >>>> measure of how the sets compare at any given point in their production. >> >>> This makes no sense. Sets are not "produced" or "generated". >>> Sets simply exist. >> >>>> Sets as sets are considered static and complete. >> >>> Correct. >> >>>> However, when talking >>>> about processes of adding and removing elements, the sets are not >>>> static, but changing with each event. >> >>> Incorrect. If we define set A as containing the elements a, b, and c, >>> then A = {a, b, c}. Period. If we then talk about adding elements d >>> and e to set A, we're not actually changing set A, but describing >>> another set, call it A2, that is the union of A and {d, e}, so >>> A2 = {a, b, c, d, e}. >> >>> Nothing is ever "added to" a set. Rather, we apply operations (union, >>> intersection, etc.) to existing sets to create new sets. We don't >>> change existing sets. >> >> Just like when we add 5 to 2 to get 7, we do not change the 5 or 2 >> to create a 7. Or when you celebrate a birthday, your age changes, >> but the number that represented your age does not change. A different >> number is used to represent your age, but the "old" number remains >> as it always ways. >> >> This idea of "changing" sets seems to be at the heart of a lot >> of people's misconceptions about set theory. >> >> Stephen > Then it's an entirely different vase, a countably infinite number of > times over and over before noon, and if there is a vase at noon at all, > it's a different vase, with different balls. That seems to fit in with > your concept of set theory, eh? Ugh. I was not talking about vases Tony, and you apparently did not understand what I was talking about. Given that you seem incapable of thinking about anything but balls and vases at the moment, I can use them as an example of what I was talking about. At the beginning, the number of balls in the vase is 0. After the first iteration, the number of balls in the vase is 9. The number of balls in the vase changes from 0 to 9, but 0 does not change into 9. 0 and 9 are immutable. The number of balls is a variable, and can have different values, but the values themselves cannot change. Similarly, at the beginning the set of balls in the vase is {}. After the first iteration, the set of balls in the vase is { 2,3,4,5,6,7,8,9,10}. The set of balls in the vase changes from {} to {2,3,4,5,6,7,8,9,10} but {} does not change into {2,3,4,5,6,7,8,9,10}. The two sets are immutable. The set of balls is a variable, and can have different values, but the values themselves cannot change. This is how mathmeticians think of numbers and sets. I imagine it is how you think about numbers. When I add 5 to 7 to get 12, I do not change a 5 or a 7 into a 12. 5, 7 and 12 are constants. The 12 I get by adding 5 and 7 is the same 12 I get when I add 6 and 6, or subtract 3 from 15. I am not making new instances of 12 when I perform these operations. Sets behave the same way. Stephen
From: stephen on 1 Nov 2006 12:38
Virgil <virgil(a)comcast.net> wrote: > In article <45481b7f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >> > David R Tribble <david(a)tribble.com> wrote: >> >> [Apologies if this duplicates previous responses] >> > >> >> Tony Orlow wrote: >> >>> I am beginning to realize just how much trouble the axiom of >> >>> extensionality is causing here. That is what you're using, here, no? The >> >>> sets are "equal" because they contain the same elements. >> > >> >> Yes, the basic definition of set equality, the '=' set operator. >> > >> >>> That gives no >> >>> measure of how the sets compare at any given point in their production. >> > >> >> This makes no sense. Sets are not "produced" or "generated". >> >> Sets simply exist. >> > >> >>> Sets as sets are considered static and complete. >> > >> >> Correct. >> > >> >>> However, when talking >> >>> about processes of adding and removing elements, the sets are not >> >>> static, but changing with each event. >> > >> >> Incorrect. If we define set A as containing the elements a, b, and c, >> >> then A = {a, b, c}. Period. If we then talk about adding elements d >> >> and e to set A, we're not actually changing set A, but describing >> >> another set, call it A2, that is the union of A and {d, e}, so >> >> A2 = {a, b, c, d, e}. >> > >> >> Nothing is ever "added to" a set. Rather, we apply operations (union, >> >> intersection, etc.) to existing sets to create new sets. We don't >> >> change existing sets. >> > >> > Just like when we add 5 to 2 to get 7, we do not change the 5 or 2 >> > to create a 7. Or when you celebrate a birthday, your age changes, >> > but the number that represented your age does not change. A different >> > number is used to represent your age, but the "old" number remains >> > as it always ways. >> > >> > This idea of "changing" sets seems to be at the heart of a lot >> > of people's misconceptions about set theory. >> > >> > Stephen >> >> Then it's an entirely different vase, a countably infinite number of >> times over and over before noon, and if there is a vase at noon at all, >> it's a different vase, with different balls. That seems to fit in with >> your concept of set theory, eh? Ugh. > The vase, when its contents are changed, remains the same vase but with > different contents. > If TO cannot tell the difference between the vase and what is in the > vase, he is more seriously handicapped than we had previously realized. I think Tony thinks the vase is a set, as opposed to containing a set. Stephen |