Uncountability of the Rationals?
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From: Tony Orlow on 11 Jun 2010 18:22 On Jun 11, 1:07 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 11, 7:15 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > On Jun 10, 2:42 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > On Jun 10, 1:35 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > and I > > > > have to retreat > > > > You retreat every time you fail to define your terminology in a non- > > > circular way, every time you fail to answer the substantive questions > > > but instead blame the actually informed people here for what you claim > > > to be their obtuseness or dishonesty, every time you fail to answer > > > the substantive questions but instead resort to things like corny "yo' > > > mamma" talk. > > No, I tend to retreat when the criticism gets too personal and harsh > > and starts to seriously erode my motivation and self esteem. > > Fair enough, but you ALSO retreat into the forms I mentioned even when > pressed in a personally neutral way. Also, you should be self-aware > enough (I doubt that you are) to notice that often YOU engender the > negatively personal aspects when you respond to people with your > customary dismissiveness, charges that others (including well informed > and talented mathematicians) are too obtuse or dishonest to understand > you, juvenile sarcasm, and ridiculous bumptiousness. > > > BTW, you > > mention a goal of set theory being to have a "size" for every set. I > > would like you consider a slightly modified goal: To exactly order > > sets according to size, and to distinguish between those that have an > > absolute size and those that do not. > > Whatever "absolute size" might mean, then fine, no one's stopping your > from pursuing such a goal. On the other hand, when you present > gobbeldygook, then you can expect that mathematically informed people > will call you on it. > > > For me, no countably infinite set > > has an absolute size, > > A countably infinite set is, by definition, 1-1 with w. If you wish to > have a system that defines as well "absolute size" then step right up > and tell it to us...primitives, axioms, definitions. > > > but only one relative to other countably > > infinite sets, especially the standard N, starting at 0. Uncountably > > infinite sets can have an absolute size in terms of zillions. Both > > kinds of infinity can be multiply tiered using infinite case induction > > on all the arithmetic functions used in bijections. > > As above... > > > Also, sometimes I retreat to rethink something, when an issue in my > > thinking is pointed out, which has happened several times, and which I > > don't regret. I welcome logical criticism. When it gets personal, I > > can get personal back, but I'm trying to behave myself. Are you? > > I've many times attempted to engage you in a personally neutral way. > It's never worked. Quickly the conversation becomes personal, often as > I steer it that way, because the elephant in the room is that you just > don't have the intellectual and personal maturity to participate in a > coherent mathematical or even math/philosophical discussion. > > But I'm always willing to try again. If you like, tell me from (a new) > beginning just what you propose. Please don't just SWAMP me with a ton > of UNDEFINED verbiage. > > What I think you fail to appreciate is how arrogant you are when you > throw a bunch of undefined verbiage at people. The effect of that is > that ONLY YOU have a definitive call in what does or does not hold in > your "mathematics". That defies the opposing intellectual standard > that a mathematical system should allow anyone to prove theorems and > devise additional definitions in it INDEPENDENT of the particular > human being who first proposed the system. Because of the way you > present matters, only you can say, from case to case, what inferences > we are to draw, because your mathematics is for the most part merely a > bunch of ideas swarming in your personal brain somewhat expressed or > sketched or alluded to with undefined terminology. > > Thus your part in the conversation is lacking in the basic > mathematical/intellectual consideration of presenting a system from > which OTHER PEOPLE may themselves draw inferences without having to > consult you as its oracle. > > And even if you are not at the stage of presenting a system, but > rather sketching some philosophical math/ontological notions, then > still what is lacking from your are (1) an understanding of the work > that's already been done so that you can put your own notions in > perspective that way, as well as that your critiques of already > presented mathematics are ill-premised, and (2) a sincere effort to > find language, even if only illustrative, that would allow other > people to approach some grasp of what it is you want in some system- > later-to-be-stated; as instead you mainly barrage us with undefined > math sounding terminology. > > So to start anew, if you wish: > > (1) What logic at this time do you propose for carrying out your > mathematics. Classical, multi-valued, para-consistent, some other > logic that you've devised? So far, I am using classical logic. If we could ever get into probabilistic logic I'd be amazed. > > (2) Do you intend for your mathematics to provide a mathematics > sufficient for the sciences? Not necessarily. Perhaps in cosmological areas. > > (3) Do you contend that your mathematical notions are in some sense > true of some ultimate (for lack of a better word) physical or abstract > world - a reality - that other mathematics (such as ZFC, > constructivist mathematics, et. al) are not true to said reality? Mostly intuitive notions, but we'll see. ELiminating proper subset equinumerosity is worthwhile. > > MoeBlee TO
From: MoeBlee on 11 Jun 2010 18:29 On Jun 11, 4:58 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > What's an invertible formula? > > More specifically (sorry) a monotonically increasing real function, So we can throw out the confusing terminology 'invertible formula', which seems to refer to formulas as opposed to functions. > > > Try it on any finite set. Now imagine applying it to the range > > > [0,omega]. So you mean f:w+ -> R and f is increasing ? Okay, all that means is that f\{<w f(w>} is a bounded inceasing sequence in R, and f(w) is an upper bound of the range of f\{<w f(w>}. Okay, what about it? MoeBlee
From: MoeBlee on 11 Jun 2010 18:38 On Jun 11, 5:22 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 11, 1:07 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > On Jun 11, 7:15 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > > On Jun 10, 2:42 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > On Jun 10, 1:35 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > and I > > > > > have to retreat > > > > > You retreat every time you fail to define your terminology in a non- > > > > circular way, every time you fail to answer the substantive questions > > > > but instead blame the actually informed people here for what you claim > > > > to be their obtuseness or dishonesty, every time you fail to answer > > > > the substantive questions but instead resort to things like corny "yo' > > > > mamma" talk. > > > No, I tend to retreat when the criticism gets too personal and harsh > > > and starts to seriously erode my motivation and self esteem. > > > Fair enough, but you ALSO retreat into the forms I mentioned even when > > pressed in a personally neutral way. Also, you should be self-aware > > enough (I doubt that you are) to notice that often YOU engender the > > negatively personal aspects when you respond to people with your > > customary dismissiveness, charges that others (including well informed > > and talented mathematicians) are too obtuse or dishonest to understand > > you, juvenile sarcasm, and ridiculous bumptiousness. > > > > BTW, you > > > mention a goal of set theory being to have a "size" for every set. I > > > would like you consider a slightly modified goal: To exactly order > > > sets according to size, and to distinguish between those that have an > > > absolute size and those that do not. > > > Whatever "absolute size" might mean, then fine, no one's stopping your > > from pursuing such a goal. On the other hand, when you present > > gobbeldygook, then you can expect that mathematically informed people > > will call you on it. > > > > For me, no countably infinite set > > > has an absolute size, > > > A countably infinite set is, by definition, 1-1 with w. If you wish to > > have a system that defines as well "absolute size" then step right up > > and tell it to us...primitives, axioms, definitions. > > > > but only one relative to other countably > > > infinite sets, especially the standard N, starting at 0. Uncountably > > > infinite sets can have an absolute size in terms of zillions. Both > > > kinds of infinity can be multiply tiered using infinite case induction > > > on all the arithmetic functions used in bijections. > > > As above... > > > > Also, sometimes I retreat to rethink something, when an issue in my > > > thinking is pointed out, which has happened several times, and which I > > > don't regret. I welcome logical criticism. When it gets personal, I > > > can get personal back, but I'm trying to behave myself. Are you? > > > I've many times attempted to engage you in a personally neutral way. > > It's never worked. Quickly the conversation becomes personal, often as > > I steer it that way, because the elephant in the room is that you just > > don't have the intellectual and personal maturity to participate in a > > coherent mathematical or even math/philosophical discussion. > > > But I'm always willing to try again. If you like, tell me from (a new) > > beginning just what you propose. Please don't just SWAMP me with a ton > > of UNDEFINED verbiage. > > > What I think you fail to appreciate is how arrogant you are when you > > throw a bunch of undefined verbiage at people. The effect of that is > > that ONLY YOU have a definitive call in what does or does not hold in > > your "mathematics". That defies the opposing intellectual standard > > that a mathematical system should allow anyone to prove theorems and > > devise additional definitions in it INDEPENDENT of the particular > > human being who first proposed the system. Because of the way you > > present matters, only you can say, from case to case, what inferences > > we are to draw, because your mathematics is for the most part merely a > > bunch of ideas swarming in your personal brain somewhat expressed or > > sketched or alluded to with undefined terminology. > > > Thus your part in the conversation is lacking in the basic > > mathematical/intellectual consideration of presenting a system from > > which OTHER PEOPLE may themselves draw inferences without having to > > consult you as its oracle. > > > And even if you are not at the stage of presenting a system, but > > rather sketching some philosophical math/ontological notions, then > > still what is lacking from your are (1) an understanding of the work > > that's already been done so that you can put your own notions in > > perspective that way, as well as that your critiques of already > > presented mathematics are ill-premised, and (2) a sincere effort to > > find language, even if only illustrative, that would allow other > > people to approach some grasp of what it is you want in some system- > > later-to-be-stated; as instead you mainly barrage us with undefined > > math sounding terminology. > > > So to start anew, if you wish: > > > (1) What logic at this time do you propose for carrying out your > > mathematics. Classical, multi-valued, para-consistent, some other > > logic that you've devised? > > So far, I am using classical logic. If we could ever get into > probabilistic logic I'd be amazed. > > > > > (2) Do you intend for your mathematics to provide a mathematics > > sufficient for the sciences? > > Not necessarily. Perhaps in cosmological areas. Then if it turns out your theory does not provide for mathematics for the sciences, what foundational theory do you propose for mathematics for the sciences? And if it turns your theory doesn't provide for mathematics for the sciences, then what reason do you give that anyone should be interested in your theory AS AN ALTERNATIVE to ZFC? You understand that if we didn't look to a foundational theory (such as ZFC) for a mathematics for the sciences then of course we'd be free to dispense with a lot what people may think to be counterintuitive. Sure, we can invent all kinds of theories with all kinds of interesting, satisfying, and intuition-friendly mathematical ontologies. But merely having an interesting, satisfying, and intuition-friendly mathematical ontology is not all we (editorial 'we') ordinarily expect from a foundational theory. > > (3) Do you contend that your mathematical notions are in some sense > > true of some ultimate (for lack of a better word) physical or abstract > > world - a reality - that other mathematics (such as ZFC, > > constructivist mathematics, et. al) are not true to said reality? > > Mostly intuitive notions, but we'll see. ELiminating proper subset > equinumerosity is worthwhile. You didn't answer that question. MoeBlee
From: MoeBlee on 11 Jun 2010 18:56 On Jun 11, 5:11 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 11, 12:45 pm, David R Tribble <da...(a)tribble.com> wrote: > > > Tony Orlow wrote: > > > I would like you consider a slightly modified goal: To exactly order > > > sets according to size, and to distinguish between those that have an > > > absolute size and those that do not. For me, no countably infinite set > > > has an absolute size, but only one relative to other countably > > > infinite sets, especially the standard N, starting at 0. > > > Wouldn't it be easier to say that some chosen countably infinite > > set has a standard "absolute" size, and then that all other such > > sets have a size that is some fraction of that size? Thus every > > set then has an "absolute" size, based on the standard set size. > > That might be easier, and would lead, as suggested, to natural > density. With the word "fraction" in scare quotes, and allowing also over the size of the "absolute" it's along the lines of ZFC cardinality. > However, that doesn't address the relations between infinite > sets that have other ratios or offsets besides the finite. Okay, but we take w as the "basic infinite size" and find that there are other sets that have larger infinite size. > It is much > more fruitful Fruitful in WHAT sense? Toward WHAT end? > to look for a unit infinity based on uncountable > infinity, which can be much more easily combined with measure and > topology. How so? (I'm skeptical that you even know what a topology is.) Anyway, in ZFC we already do have a least uncountably infinite size. Take that as the "basic infinite size" if you like. Then w is one infinite size less. > > For convenience, you'd want to choose the "largest" countably > > infinite set for the standard "absolute" set size. This would > > probably be N, since it contains all of the countable naturals. > > (Another logical choice would be Z.) Given this, you could then > > say that every countably infinite set does indeed have an > > "absolute" set size.\ I'd take the least (where 'least' means is a subset of all others). I don't know how w would be reckoned as the largest. > But, for me, they do not. If N starts at the first location with 1, Then N is w\{0}. > and then the second with 2, etc, then the nth natural is n, and unless > a set has n elements, there is no nth one. If there are omega > naturals, then there is an omegath, You simply POSIT that. It's not the case in ordinary mathematics, so if you want it, then you need axioms to prove it from. > which by the order=value > definition of that set must equal omega. However omega cannot exist > within the set. Consider this. Are the following two statements > logically equivalent? > > AneN n<omega > > ~EneN n>=omega No, they are not LOGICALLY equivalent. They are equivalent in certain THEORIES, such as ZFC. > The first says there is no ntaural that can be the size of the entire > set of naturals. That's true. It entails what you just said; I wouldn't say that is what it says in itself. > The second says that there is no n in N that is greater than or equal > to this "size". Okay, I'll play along with that. > Does either imply, actually, that such a size exists? No. We have an AXIOM that entails that there is a set that has as members all and only the positive natural numbers (what you call 'N'). > Both assert > that no natural number will suffice. However, this is not to say that > there exists ANY number which does. The formulas you mentioned don't entail that there is such an ordinal number, true. But we do prove from our AXIOMS that there exists such an ordinal number. > Logically, you must concede, > neither statement proves that omega exists as a number. There's really nothing to concede. It's never been at issue. > I am not > obligated to imagine such a number, or give it any credibility. Of course you're not. You're not obligated to imagine anything at all. > When > it comes to the countably infinite, we can only look at this as some > kind of limit, It is a "limit" in a certain special sense; it is a limit ordinal. So what? > and for most discriminating results, consider them as > Big-O measures or something similar. Too bad I don't have my dictionary of Orlowism to look up 'Big-o measure'. Anyway, in all your verbiage I still don't see a substantive point other than, pretty much, this: The existence of a set of all the positive natural numbers doesn't follow from the formulas you mentioned. Yes, we agree. MoeBlee
From: MoeBlee on 11 Jun 2010 19:04
On Jun 11, 5:15 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 11, 12:44 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > Tony Orlow <t...(a)lightlink.com> writes: > > > On Jun 11, 12:21 am, David R Tribble <da...(a)tribble.com> wrote: > > >> Tony Orlow wrote: > > >> > For, "none > > >> > shall drive us from the Garden which Cantor has created for us". If it > > >> > doesn't produce fruit, it's time to plant a new bed, or at least > > >> > fertilize. > > > >> Well, it's done pretty well for the last 100+ years. Have any > > >> of your ideas produced any results of consequence yet? > > > >> I'm still waiting to hear what bigulosity/IFR says about the size > > >> of the set of natural squares. Any results yet? > > > > I already did that. According to IFR it's sqrt(omega), which according > > > to ICI is less than omega. > > > So, it is a(n ordinal) number x such that x * x = omega, is that right? > > Ordinals have nothing to do with my theory. I've called the von > Neumann ordinals "schlock" for years. You should know that, at the > very least. But w satisfies the definition of 'an ordinal number'. So whether you think they are schlock or whatever, you're still talking about them in your own formulations. Also, you say you don't object to ZFC except as it has been "extended" (whatever that means). The ordinals come right out of the axioms and definitions used with ZFC. It really is not what you're on about. And and ordinal is simply a set that is epsilon-transitive and well ordered by epsilon. I don't know why you think that such a thing is "schlock". But I think it's LIMIT ordinals that you object to. Then you do object to ZFC, as ZFC proves the existence of limit ordinals. So, get some axioms that prove that there don't exist limit ordinals or that leave the matter undecided. > > As well, one can prove that omega is the least infinite ordinal (where > > "least" is defined in terms of cardinality, of course). That is, if > > alpha is an infinite ordinal, then there is an injection from omega to > > alpha. Is this contrary to your claims? You sometimes say that you > > don't think omega is the smallest infinite ordinal, but I'm not sure > > what you mean when you say that. > > I don't use the word "ordinal" in my arguments (sure, go find one > mention from 1996 or whatever). I say there are a wide spectrum of > countable and uncountable infiniites, given the right techniques. I say that given the right techniques we can find a wide spectrum of delicious fruits and vegtables that no one in the world has tasted. Too bad I can't make my "techniques" clear to any other human being. MoeBlee |