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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 n

> 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.
  - RiderOfGiraffes 16y ago
    I don't have time now to go over this in detail, but again, you're not communicating things to me clearly. However, >> 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! No, it's no argument, it's an attempt to give you the right intuition. It is what's true, and I'm trying to help you visualise it. >> 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. Not when you've worked for sufficiently long with these things. As you work with them - not just learn the definitions and a few standard proofs - you start to gain an intuition for what's really going on. People are trying to help you gain that intuition by telling you these things, and you're simply rejecting it and arguing from your existing intuition. > 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:) My intuition isn't just based on a few facts, it's based on working with these things for years, and the intuition developed. And the cardinality of the reals isn't just "a bit bigger". Cantor's standard proof may have left you with that feeling, but go back to the proof I gave earlier. Start with a countable collection of reals. The standard diagonal argument gives you one that's missing. Now look at just how many ways you can generate new reals from that one list. You can take the first from the second and the second fromthe first, 3rd from 4th and 4th from 3rd, etc. In fact, any permutation of the infinite list (with certain restrictions as to hitting everything) gives you missing elements. It's not just "a slippery bit bigger" (a term for which you have given me no efinition and no intuition), it's enormously bigger. The one intuition you come back to is the "in between" bit. Again I give you the example of the evens. Surely since between every pair of even numbers there's an integer, that means there must be more integers? No, it doesn't, and it's just like that. You say that because there are infinitely many rationals between each pair of reals then there must be a similar number of reals (ok, maybe "a slippery bit more"). Well, no, that's not true. It just isn't like that. In short, your existing intuition is leading you astray, you haven't worked enough with the infinities and their applications to develop an appropriate intuition, and you're rejecting every attempt to help you gain the right intuition. So I'll stop there until you can help me understand your problem better.

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.
- RiderOfGiraffes 16y ago
  I don't have time now to go over this in detail, but again, you're not communicating things to me clearly. However, >> 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! No, it's no argument, it's an attempt to give you the right intuition. It is what's true, and I'm trying to help you visualise it. >> 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. Not when you've worked for sufficiently long with these things. As you work with them - not just learn the definitions and a few standard proofs - you start to gain an intuition for what's really going on. People are trying to help you gain that intuition by telling you these things, and you're simply rejecting it and arguing from your existing intuition. > 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:) My intuition isn't just based on a few facts, it's based on working with these things for years, and the intuition developed. And the cardinality of the reals isn't just "a bit bigger". Cantor's standard proof may have left you with that feeling, but go back to the proof I gave earlier. Start with a countable collection of reals. The standard diagonal argument gives you one that's missing. Now look at just how many ways you can generate new reals from that one list. You can take the first from the second and the second fromthe first, 3rd from 4th and 4th from 3rd, etc. In fact, any permutation of the infinite list (with certain restrictions as to hitting everything) gives you missing elements. It's not just "a slippery bit bigger" (a term for which you have given me no efinition and no intuition), it's enormously bigger. The one intuition you come back to is the "in between" bit. Again I give you the example of the evens. Surely since between every pair of even numbers there's an integer, that means there must be more integers? No, it doesn't, and it's just like that. You say that because there are infinitely many rationals between each pair of reals then there must be a similar number of reals (ok, maybe "a slippery bit more"). Well, no, that's not true. It just isn't like that. In short, your existing intuition is leading you astray, you haven't worked enough with the infinities and their applications to develop an appropriate intuition, and you're rejecting every attempt to help you gain the right intuition. So I'll stop there until you can help me understand your problem better.