An uncountable countable set
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From: cbrown on 1 Nov 2006 14:34 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. > But we only let a finite amount of water out at any time; so at any time before noon we still have an /infinite/ amount of water remaining. Suddenly, at noon, the water is all gone, even though no water actually leaves the vase /at/ noon. How is it that he can accept this scenario, but not the main one? I think it's because in the former case, his intuition is that the amount of water in the vase is somehow decreasing (this accords with his idea that |N| > |N\{1}|); so it can become empty at noon. In the latter case, the amount of water is clearly increasing at each "iteration"; so it must continue to fill up, and thus cannot be empty at noon. He chooses not to consider whether this is consistent with the fact that by his own lights, the amount of water in the vase in the second scenario at any time t is always less than the amount of water in the vase at that same time in the former scenario. So, by his "infinite induction", if b_n < a_n for all n, then lim b_n < lim a_n; and we have a contradiction. But that is not a problem; because "those are different situations - it's an obvious obfuscation of the original problem". Consistency - it's not everyone's cup of tea! Cheers - Chas
From: Lester Zick on 1 Nov 2006 14:50 On Tue, 31 Oct 2006 10:30:08 -0500, Tony Orlow <tony(a)lightlink.com> wrote: [. . .] (Tony, since we have an already established audience on this thread I'm piggybacking what amounts to a new line of reasoning to your post to which I've already replied in detail. I hope the following explains exactly the origins and significance of mathematical and arithmetic infinities in purely mechanical terms.) Real Theory ~v~~ I'm now of the opinion that there is a specific reason why modern math and set analysis are wrong in fundamental mechanical terms. The difficulty has to do with what are thought of as real number lines and their supposed characteristics as lines. In effect if we use the Peano axioms and the suc( ) axiom to generate the naturals, we lock into a system of straight line segments which never correspond to curves and transcendental numbers and infinities drawn in terms of those curves. In other words it is quite possible to generate straight lines in such terms but there can never be an exact equivalence between those lines and any kind of transcendental infinity. Thus we can treat arithmetic infinites which exist in terms of infinitesimal subdivision of straight lines such as irrationals like the square root of two but not those which exist in terms of transcendental infinities such as pi. On the other hand, however, we can proceed in the opposite direction quite easily by generating straight lines as tangents through Newton's calculus and his method of tangents. And in so doing we can develop all possible reals through the mere assumption of curves instead of straight line segments and infinitesimal subdivision instead of Peano axioms and the suc( ) axiom. The problem is and always has been that mathematics in general is not arithmetic in particular. And we can always generate straight lines in terms of curves through tangency but not vice versa because in terms of form there is only one straight line but infinite kinds of curves to which there are straight line tangents and we can't proceed from any straight line tangent backwards to any specific curve. In effect then arithmetic theoreticians have straight lines but they cannot deduce curves from those straight lines and are forced to imagine such collateral forms of infinity either superimposed on straight lines themselves, such as imaginary real number lines, or in some other group altogether. But in neither case can they deduce the existence and properties of such transcendental infinities from the existence and properties of straight lines produced by arithmetic axioms except through approximation with straight line segments. Mechanical Implications ~v~~ Although our primary interest here is mathematical there is a good deal more significance than mere conventional mathematics would suggest or imply. We and all forms of being operate and think in terms of curves or at least in non straight line forms through tautological negation which is demonstrably true. However at the conceptual level we communicate with one another through straight linear forms. This is only true and possible because there is only the one form of straight line but an unlimited number of curves. Thus each of us operating at the level of abstract thought has to reduce curvilinear tautological results to straight line tangents in order to compare to and communicate thoughts of one ontological individual with those of another. And this process is exact through tangency with those curvilinear tautological results. However then we are left to ponder the origin of those exact results because we cannot reverse the process in exhaustive mechanical terms to determine with which curvilinear tautological form the tangency originated. ~v~~
From: cbrown on 1 Nov 2006 15:11 David Marcus wrote: > Randy Poe wrote: > > Tony Orlow wrote: > > > Randy Poe wrote: > > > > Tony Orlow wrote: > > > >> Randy Poe wrote: > > > >>> Do you deny me the ability to create a set of variables > > > >>> t_n, n = 1, 2, ...? Why do vases have to come into it? > > > >> I thought we were trying to formulate the problem. > > > > > > > > No, we (some of us) are trying to formulate a completely > > > > different problem, with balls and vases (possibly even > > > > times) explicitly removed so that other aspects can be examined. > > > > > > > > Yet you keep trying to put balls and vases back in, after being told > > > > that they are not present in the new problem. > > > > > > I am pointing out that your formulation doesn't match the original > > > problem. > > > > It's a new problem. > > > > Are you capable of contemplating a second problem, > > throwing away balls and vases and starting from scratch, > > asking different questions about a different problem? > > That's a good question. I was thinking of asking Tony the same thing. If > Tony can only discuss one problem, it does limit the discussion. Best we > know that up front. > Tony can discuss many problems; he just doesn't see the relevance of this line of reasoning. Tony's approach is to examine a problem, assume some plausible conclusion in accord with his intuitions, and from this deduce what the assumptions in the problem actually mean so that they correspond to this conclusion. "Because the vase cannot be empty at noon, there are infinite naturals". >From this point of view, different problems can thus yield a different set of assumptions about "what happens". The fact that the assumptions he deduces in one case may contradict the assumptions he deduces in another case is largely irrelevant; because he doesn't base his arguments on assumptions, he bases them on the "obvious" conclusions. In other words, to change the problem in any way and then try to assert that the assumptions in both problems should be the same is a T-non-sequitor. 'S a deliberate obfuscation of the original problem. :) Cheers - Chas
From: David Marcus on 1 Nov 2006 15:46 stephen(a)nomail.com wrote: > 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. That could very well be. It is always a good idea to have a clear separation between English and Mathematics. If something changes, then that's usually a clue that when you go from English to Mathematics, you should stick a function in there. -- David Marcus
From: David Marcus on 1 Nov 2006 16:00
cbrown(a)cbrownsystems.com wrote: > David Marcus wrote: > > Mike Kelly wrote: > > > 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. > > But we only let a finite amount of water out at any time; so at any > time before noon we still have an /infinite/ amount of water remaining. > Suddenly, at noon, the water is all gone, even though no water actually > leaves the vase /at/ noon. > > How is it that he can accept this scenario, but not the main one? > > I think it's because in the former case, his intuition is that the > amount of water in the vase is somehow decreasing (this accords with > his idea that |N| > |N\{1}|); so it can become empty at noon. In the > latter case, the amount of water is clearly increasing at each > "iteration"; so it must continue to fill up, and thus cannot be empty > at noon. Yes, but I think it is simpler. You are still trying to apply logic to the mental picture. When the mental picture differs from the math, you rely on the math to draw conclusions. This is not the way most people think. Most people don't use logic at all. If you have a mental picture of a vase that is full of water and you start to let water drain out, then the amount of water in the vase is obviously decreasing. The fact that the mathematics says that the amount of water in the vase before noon is always infinite just doesn't agree with the mental picture. You can't really have a mental picture of an infinite vase full of an infinite amount of water, so I doubt anyone really has such a picture, unless they've overlaid the math onto their mental picture, and learned to think mathematically/logically. > He chooses not to consider whether this is consistent with the fact > that by his own lights, the amount of water in the vase in the second > scenario at any time t is always less than the amount of water in the > vase at that same time in the former scenario. So, by his "infinite > induction", if b_n < a_n for all n, then lim b_n < lim a_n; and we have > a contradiction. He's not using logic at all. He probably doesn't agree that b_n < a_n. After all, the mental pictures don't show this. In one case, you've got a vase filling with water. In the other, you've got a full vase that is emptying. Clearly the vase that is emptying can't always have more water than the vase that is filling! > But that is not a problem; because "those are different situations - > it's an obvious obfuscation of the original problem". > > Consistency - it's not everyone's cup of tea! If you aren't using logic, then you don't even understand what consistency is. On the other hand, I think he is being consistent with the mental pictures. The problem is that the problem is intentionally non-physical, so you can't create an accurate mental picture without embracing the mathematics. -- David Marcus |