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The reals are not Well-Ordered under the usual ordering, and they arenot Well Ordered at all. Certainly if you create a Well Ordering of the reals then it's no

The reals are not Well-Ordered under the usual ordering, and they arenot Well Ordered at all. Certainly if you create a Well Ordering of the reals then it's not the usual ordering. And you are free to work with your own intuition, but it's very, very likely that your intuition will be inconsistent with other axioms that you accept to be true. Once you have worked with them enough there is nothing bizarre about every two reals having coutnably many rationals between them, and every two rationals having uncountably many reals between them. I just think that if I have "almost all" of the stuff, you shouldn't be able to have an infinite amount of stuff between every given two pieces of my stuff! This strikes me as very strange. If the rationals (or algebraics) are countable, then "almost all" of the reals are not rationals. Why should it not be that between any two transcendentals (of which there are uncountably many) there are "a few" (by which I mean only countably many) rationals? I think the point to make is that for centuries, perhaps millennia, people grappled unsuccessfully with the ideas of infinite cardinalities. Many, many mistakes were made, and finally there is now a coherent, consistent way of working with them. To say they don't match your intuition is simply to say that your intuition is at odds with the centuries of work done by people who devoted their lives to working on it. In short, you are welcome to your intuitions, but I suspect they are hindering you substantially from understanding things better. You can develop better intuition by working with these ideas "properly." But that's harder than just saying "My intuition doesn't agree with what the mathematicians say."

WilliamLP 16y ago
> The reals are not Well-Ordered under the usual ordering, and they arenot Well Ordered at all. Certainly if you create a Well Ordering of the reals then it's not the usual ordering. Again, you assume I don't have a basic math education! I do!! I know about choice and Zorn's and stuff like that! Obviously the usual ordering of the reals is not a well-ordering. > intuition will be inconsistent with other axioms that you accept to be true. Like? > Why should it not be that between any two transcendentals (of which there are uncountably many) there are "a few" (by which I mean only countably many) rationals? Your intuitive argument here strikes me as circular. Yes there are uncountable reals between two rationals, and countable rationals between two reals. But that only implies "almost all" if you accept that "uncountable" already means "almost all" a priori! Where does my intuition break down if I think of it as a "slippery little bit" more? Please don't assume I don't have basic knowledge of undergrad pure math in your answer, as mathematicians always do.
- RiderOfGiraffes 16y ago
  OK, in short, I don't understand your position. It seems indistinguishable from someone who doesn't understand anything about the infinities, and then you assert that you have a degree in math. OK. Let me quote you: Where I part ways with mathematicians is the intuition we should ascribe to Cantor's argument. They say that "almost all" real numbers are irrational. As you know, that claim does not arise from Cantor's argument at all, it comes from measure theory. Cantor's argument about the uncountability of the reals (or of the set of subsets of a countable set, or the fact that the power set is always strictly bigger) says nothing about "almost all." If you want to talk about "almost all" then you have to go to measure theory. Having said that, certainly if you color all the rationals red, and all the irrationals blue, then there are more blue than reds, and the amount by which it's more is a lot. What Cantor's argument says is that "at least one" is irrational. No. Cantor's argument says that for every countable set of reals, at least one is missing. Cantor's argument actually says nothing about rationals or irrationals. I know the arguments about measure and probability and such, OK - I'll take that as a true statement. ... there is also the fact that for every two real numbers, there is an infinity of rational numbers in between! So? What do you think is the problem with that? Why does this cause you difficulty? In particular, it's only a small infinity. Between any two rationals there is a bigger infinity of irrationals. Your problem seems to be that you are trying to visulaise this in terms of finite collections and finding inconsistencies/contradictions. It's like being confused that the set of even numbers is the same size as the set of integers, when it should be "half the size." It's pretty much the same type of confusion, but at the next level up. So that makes me (personally, not mathematically) not like the "almost all" intuition, instead of an intuition of there always being at least one more, in a particularly slippery way. This doesn't make any sense to me - it communicates no knowledge or understanding of your position. You will need to expand it substantially before I can try to help you adjust your intuition to be more in tune with mainstream mathematics.
  - WilliamLP 16y ago
    > If you want to talk about "almost all" then you have to go to measure theory. From my understanding, "almost all" is somewhat imprecise and can mean many different things even in math, e.g. all but countably many, all but for a set with Lesbegue measure 0, with the precise meaning either stated explicitly or depending on the context. So if it's "almost all" by definition, that fine, that's your definition. That's math. That's why I stress, again, that I'm not talking about precise math. I do feel that mathematicians tend to use "almost all" in the common sense though as well when talking about uncountable sets, where it isn't precise, and isn't well defined, and actually isn't entirely consistent with our usual intuition. That is where I disagree. > Having said that, certainly if you color all the rationals red, and all the irrationals blue, then there are more blue than reds, and the amount by which it's more is a lot. Again, circular argument! If I say the difference between uncountable and countable isn't "a lot", but actually more like "a little", it doesn't follow. One way of stating my proposed intuition would be that there is a big difference between finite and infinite, and a "slippery little" between countable and uncountable. (Where they nonetheless have some important theoretical distinctions, including the idea of measure.) Against this intuition is probability (whatever choosing a real number at random actually means...) But for it is the idea of the symmetry between these infinities, that there is an infinity of one between the other and vice versa. > Your problem seems to be that you are trying to visualise this in terms of finite collections and finding inconsistencies/contradictions. Yes, when you start using imprecise terms like "almost all", you're talking about intuition, and our intuitions are of course based on finite sets. I think your problem is that you think that one set of math facts (including probability, measure, lack of a bijection) that is sort of analogous to an intuition of finite sets, is the preferred viewpoint, over another set of math facts (infinity of rationals between reals) in shaping your intuition. To your credit, all mathematicians seem to be on your side:) I'm looking for the killer fact that destroys my intuition, but nobody has ever been able to give it to me beyond the standard undergrad stuff that everyone who knows anything about math knows.

WilliamLP 16y ago

> The reals are not Well-Ordered under the usual ordering, and they arenot Well Ordered at all. Certainly if you create a Well Ordering of the reals then it's not the usual ordering. Again, you assume I don't have a basic math education! I do!! I know about choice and Zorn's and stuff like that! Obviously the usual ordering of the reals is not a well-ordering. > intuition will be inconsistent with other axioms that you accept to be true. Like? > Why should it not be that between any two transcendentals (of which there are uncountably many) there are "a few" (by which I mean only countably many) rationals? Your intuitive argument here strikes me as circular. Yes there are uncountable reals between two rationals, and countable rationals between two reals. But that only implies "almost all" if you accept that "uncountable" already means "almost all" a priori! Where does my intuition break down if I think of it as a "slippery little bit" more? Please don't assume I don't have basic knowledge of undergrad pure math in your answer, as mathematicians always do.

RiderOfGiraffes 16y ago
OK, in short, I don't understand your position. It seems indistinguishable from someone who doesn't understand anything about the infinities, and then you assert that you have a degree in math. OK. Let me quote you: Where I part ways with mathematicians is the intuition we should ascribe to Cantor's argument. They say that "almost all" real numbers are irrational. As you know, that claim does not arise from Cantor's argument at all, it comes from measure theory. Cantor's argument about the uncountability of the reals (or of the set of subsets of a countable set, or the fact that the power set is always strictly bigger) says nothing about "almost all." If you want to talk about "almost all" then you have to go to measure theory. Having said that, certainly if you color all the rationals red, and all the irrationals blue, then there are more blue than reds, and the amount by which it's more is a lot. What Cantor's argument says is that "at least one" is irrational. No. Cantor's argument says that for every countable set of reals, at least one is missing. Cantor's argument actually says nothing about rationals or irrationals. I know the arguments about measure and probability and such, OK - I'll take that as a true statement. ... there is also the fact that for every two real numbers, there is an infinity of rational numbers in between! So? What do you think is the problem with that? Why does this cause you difficulty? In particular, it's only a small infinity. Between any two rationals there is a bigger infinity of irrationals. Your problem seems to be that you are trying to visulaise this in terms of finite collections and finding inconsistencies/contradictions. It's like being confused that the set of even numbers is the same size as the set of integers, when it should be "half the size." It's pretty much the same type of confusion, but at the next level up. So that makes me (personally, not mathematically) not like the "almost all" intuition, instead of an intuition of there always being at least one more, in a particularly slippery way. This doesn't make any sense to me - it communicates no knowledge or understanding of your position. You will need to expand it substantially before I can try to help you adjust your intuition to be more in tune with mainstream mathematics.
- WilliamLP 16y ago
  > If you want to talk about "almost all" then you have to go to measure theory. From my understanding, "almost all" is somewhat imprecise and can mean many different things even in math, e.g. all but countably many, all but for a set with Lesbegue measure 0, with the precise meaning either stated explicitly or depending on the context. So if it's "almost all" by definition, that fine, that's your definition. That's math. That's why I stress, again, that I'm not talking about precise math. I do feel that mathematicians tend to use "almost all" in the common sense though as well when talking about uncountable sets, where it isn't precise, and isn't well defined, and actually isn't entirely consistent with our usual intuition. That is where I disagree. > Having said that, certainly if you color all the rationals red, and all the irrationals blue, then there are more blue than reds, and the amount by which it's more is a lot. Again, circular argument! If I say the difference between uncountable and countable isn't "a lot", but actually more like "a little", it doesn't follow. One way of stating my proposed intuition would be that there is a big difference between finite and infinite, and a "slippery little" between countable and uncountable. (Where they nonetheless have some important theoretical distinctions, including the idea of measure.) Against this intuition is probability (whatever choosing a real number at random actually means...) But for it is the idea of the symmetry between these infinities, that there is an infinity of one between the other and vice versa. > Your problem seems to be that you are trying to visualise this in terms of finite collections and finding inconsistencies/contradictions. Yes, when you start using imprecise terms like "almost all", you're talking about intuition, and our intuitions are of course based on finite sets. I think your problem is that you think that one set of math facts (including probability, measure, lack of a bijection) that is sort of analogous to an intuition of finite sets, is the preferred viewpoint, over another set of math facts (infinity of rationals between reals) in shaping your intuition. To your credit, all mathematicians seem to be on your side:) I'm looking for the killer fact that destroys my intuition, but nobody has ever been able to give it to me beyond the standard undergrad stuff that everyone who knows anything about math knows.