An uncountable countable set
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From: Tony Orlow on 31 Oct 2006 22:56 David R Tribble 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. > Which views the sets as static. >> 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. > Then you cannot use sets to represent the changing number of balls in the vase due to the sequence of events. >> Sets as sets are considered static and complete. > > Correct. > So they don't represent sums due to sequences of additions and subtractions. >> However, when talking >> about processes of adding and removing elements, the sets are not >> static, but changing with each event. > > Incorrect. Then sets don't apply. 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}. Then sets don't apply. > > 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. Then what's in the vase is not a set, and set theory does not apply. > >> When speaking about what is in the >> set at time t, use a function for that sum on t, assume t is continuous, >> and check the limit as t->0. Then you won't run into silly paradoxes and >> unicorns. > > That would be using the wrong nomenclature. We don't talk about > what set A contains at any particular time t. We can talk about the > sequence of sets A_1, A_2, A_3, etc. We can also talk about the > union of the entire sequence of sets as set A, if we like. Or the > intersection, or whatever. But every set we're talking about is > "static", unchanging, once it (the members it contains) is defined. > Then talk about the sequence of events each of which changes the number of balls, and leaves sets out of it. They don't describe sequences.
From: Tony Orlow on 31 Oct 2006 22:59 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.
From: Virgil on 31 Oct 2006 23:10 In article <454819f4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Are you capable of contemplating a second problem, > > throwing away balls and vases and starting from scratch, > > asking different questions about a different problem? > > > > - Randy > > > > I am capable, if I have a reason. Does it shed light on the original > problem, or is it simply a distraction from the logical question about > the problem at hand? I'm not really interested in pointless distractions. At it is an attempt to make TO see truths he would rather deny, he is not interested.
From: Virgil on 31 Oct 2006 23:23 In article <45481ae5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble 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. > > > > Which views the sets as static. Which sets are. > > >> That gives no > >> measure of how the sets compare at any given point in their production. At different times in the "production" one has different sets. The set one has at any particular time is a function of the particular time. Different times may produce different sets. > Then you cannot use sets to represent the changing number of balls in > the vase due to the sequence of events. You can use a set to represent the set of balls in the vase during any interval between changes, or after all changes are complete. > > 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}. > > Then sets don't apply. That TO is too stupid to know how to apply sets does not prevent them from applying. > > > > > 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. > > Then what's in the vase is not a set, and set theory does not apply. What is in the vase at any instant during which no change is occurring is a set of balls. > > > > >> When speaking about what is in the > >> set at time t, use a function for that sum on t, assume t is continuous, > >> and check the limit as t->0. Then you won't run into silly paradoxes and > >> unicorns. > > > > That would be using the wrong nomenclature. We don't talk about > > what set A contains at any particular time t. We can talk about the > > sequence of sets A_1, A_2, A_3, etc. We can also talk about the > > union of the entire sequence of sets as set A, if we like. Or the > > intersection, or whatever. But every set we're talking about is > > "static", unchanging, once it (the members it contains) is defined. > > > > Then talk about the sequence of events each of which changes the number > of balls, and leaves sets out of it. They don't describe sequences. Those "sequences" are sequences of sets. What TO cannot stand is that when one uses sets as they should be used, the result is clearly counter to the nonsense TO is trying to sell.
From: Virgil on 31 Oct 2006 23:28
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. |