An uncountable countable set
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From: Tony Orlow on 15 Sep 2006 15:34 David R Tribble wrote: > Tony Orlow wrote: >>> Yes, I am including infinite values on the number line, since it's >>> "infinitely long". > > David R Tribble wrote: >>> Yet another thing you have to define or prove. Where do these >>> "infinite values" appear on your real number line? > > Tony Orlow wrote: >> Further from 0 than any finite number. > > Then how are they "on" the same "line"? > By trichotomy. For all x ad y on the line, either x>y, x=y or x<y. That's what makes a line in concept. > > David R Tribble wrote: >>> Are these infinite values connected (in the mathematical sense) to >>> the finite values on the line? If not, how is it a "line"? > > Tony Orlow wrote: >> Yes, they are on the line, with all finite values between them and 0. > > That's not what "connected" means. If they are not connected, > they are simply two separate sets, which means it is more correct > to visualize them as two unconnected (but "ordered") "lines". > That is like saying the starting point of a line segment is disconnected from the rest of the line. All finite concatenations of points yields what appears to be a point - there is no finite extension. It is not until an infinite number of points are concatenated that it becomes finitely measurable. There is no greatest finite number of points with no measure, and there is no smallest infinite number of points with measure, but the two types of sequence certainly lie along the same spectrum of concept. In the same way, the infinites are simply beyond the finites, without a specific bound like the fictitious aleph_0. > > Tony Orlow wrote: >>> You are mistakenly applying >>> the standard cardinalistic fact that the number or reals in [0,1] is the >>> same as the number of all reals. That is false in my system. > > David R Tribble wrote: >>> Since it's rather easily proved true in standard theory, you need >>> to provide that missing proof of yours showing that it's false in >>> your system. For instance, you have to demonstrate that there >>> are more reals in (0,2] than in (0,1] while showing that both >>> intervals are dense. Good luck with that. (Hint: assume that >>> there are only a finite number of reals in any interval.) > > Tony Orlow wrote: >>> Given the axiomatic statement that there are Big'un reals in every unit >>> interval, that 2*Big'un>Big'un, and that (0,1] U (1,2] is (0,2], that's >>> done. > > David R Tribble wrote: >>> Without proving the second part about denseness. So we can >>> conclude that you're assuming that there are only a finite number >>> of reals in any interval (as I suspected). > > Tony Orlow wrote: >> No, Big'un is an infinite unit. Given denseness derived from the axiom >> of internal infinity, ExeR EzeR x<z -> EyeR x<y<z, there are an infinite >> number of reals in the interval. > > You still have to put the two together, proving that the number of > reals in [0,2] is more than the number of reals in [0,1] AND at the > same time that both intervals are dense in the reals. You have > done no such thing so far. Just stating two axioms is not the same > as proving that they are consistent with each other. The axiom of internal infinity applies equally well to all nonzero intervals in the reals, so denseness of both intervals is established. IFR can be applied using the naturals as a base set, which can map to the reals in (0,1] using f(x)=x/Big'un, yielding Big'un reals per unit interval, and a direct relation between element (point) count and measure. :) > > Your axiom of denseness, by itself, can be used to show that > any two real intervals have the same number of points. Your > definition of Bigun contradicts this, so you have to show how both > can be true at the same time. > There is no axiom or implication of mine that a simple 1-1 correlation means equal count for infinite sets. In fact, IFR contradicts that notion. If you think anything I have suggested supports your assertion that being infinitely divisible means being of equal size, please state where I said any such thing. I don't think I did. > > David R Tribble wrote: >>> Unless you have a definition or axiom about denseness that >>> is consistent with your theorem above? (The standard accepted >>> definition of "dense" is not, of course.) > > Tony Orlow wrote: >> I don't understand why you assume Big'un is finite, when it's an >> infinite unit. > > Because your statement about Bigun can't be consistent with > your axiom of denseness if it's not finite. But please, show us > your proof. > That's entirely backwards. The two statements imply each other. Given that the axiom of internal infinity is applied using nested even intervals (divisions by 2), we can easily see that at the nth iteration of division, each subinterval will be 1/2^n of the original interval, which is a finite fraction of a finite interval, and therefore finite, and further divisible. Therefore, if all points in the interval are to be identified as endpoints of subintervals resulting from subdivisions, the number of subdivisions must be greater than any finite number, or infinite. It's quite obvious that there are more than any finite number of reals in (0,1].
From: stephen on 15 Sep 2006 15:30 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> Representation is number. There is no difference. Numerals have no >> "soul". > Whew! I've never heard someone expressing this fact so lucidly! > Han de Bruijn So 3/2 is a different number than 6/4? The representations differ, and if there is no difference between representation and number, then different representations must imply different numbers. For that matter 6 birds is a different representation of 6 than 6 cars. And 6 birds is a different representation of 6 than 6 different birds? How many 6's are there? Stephen
From: Virgil on 15 Sep 2006 15:45 In article <450afba4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > Tony Orlow wrote: > >> You don't have it quite right. While there are countably infinite bit > >> strings in the power set of the naturals which don't correspond to any > >> standard finite natural, I contend that these strings cannot represent > >> infinite values, given that all bit positions are finite, and that they > >> qualify as finite naturals, ... What TO contends is false. > > > > They can't be finite naturals if they have an infinite number of > > nonzero digits. That would mean that s=1+2+4+8+... is a finite > > quantity and does not diverge, which is obviously false. > > It diverges as n->oo, but remains finite as long as n is finite. If it > never reaches that infinite limit for n, then 2^n-1 is never infinite > either. There are, in standard mathematics, definitions of convergence and divergence for infinite sequences and series. According to those definitions, unless one has convergence, there is no such thing as a "value" for the sequence or series. Thus, the infinite series 1 + 2 + 4 + ... does not have any value at all. > > > > They also cannot represent infinite values, since none of their finite > bit positions allows it, and that's all they have. Unless such binary strings are limited to finitely many non-zero digits, they are divergent series and thus have no real value, much less any natural value. > So, what do they > represent, then? Divergent series, which have no 'value' at all. divergent One can construct the set of all functions from N to {0,1}, and even define addition and multiplication on that set, but only a miniscule subset, those with finite carrier, will look anything like the set of naturals with its arithmetic. And because of the minimal property of the set of naturals among inductive sets, nothing larger than the set of standard naturals can be a set of naturals.
From: Tony Orlow on 15 Sep 2006 15:48 Mike Kelly wrote: > Tony Orlow wrote: >> Mike Kelly wrote: >>> Tony Orlow wrote: >>>> Mike Kelly wrote: >>>>> Tony Orlow wrote: >>>>>> Mike Kelly wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> Virgil wrote: >>>>>>>>> In article <44fe2642(a)news2.lightlink.com>, >>>>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>> >>>>>>>>>> Virgil wrote: >>>>>>>>>>> In article <44fd9eba(a)news2.lightlink.com>, >>>>>>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>>>> >>>>>>>>>>>> Dik T. Winter wrote: >>>>>>>>>>>>> In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> >>>>>>>>>>>>> writes: >>>>>>>>>>>>> Your axiom uses things that are not defined. What is the *meaning* of >>>>>>>>>>>>> "x<z"? >>>>>>>>>>>> Geometrically it means that x is left of z on the number line. >>>>>>>>>>> And for someone standing on the other side of the number line would x be >>>>>>>>>>> on the right of z? >>>>>>>>>>> >>>>>>>>>>> And does the line stay horizontal as one moves around earth? Which way >>>>>>>>>>> is larger if the line ever goes vertical. And how does the "larger" work >>>>>>>>>>> at antipodes? >>>>>>>>>>> >>>>>>>>>> Silly questions. >>>>>>>>> In response to a silly definition. >>>>>>>>>>>> It means >>>>>>>>>>>> A y (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y). That says about all it >>>>>>>>>>>> needs to, wouldn't you say? >>>>>>>>>>> Not hardly. >>>>>>>>>>> A y (y = x) v (y = z) v (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y) >>>>>>>>>>> is a bit better but still insufficient. >>>>>>>>>> True, I should have specified y<>x and y<>z. I guess it's usually done >>>>>>>>>> using <= for this reason, eh? >>>>>>>>>> >>>>>>>>>>>>> > > That is not a definition, because it makes no sense. "The set of >>>>>>>>>>>>> > > naturals >>>>>>>>>>>>> > > is as large as every natural"? >>>>>>>>>>>>> > >>>>>>>>>>>>> > It is not larger than all naturals >>>>>>>>>>>>> >>>>>>>>>>>>> That is something completely different again. >>>>>>>>>>>> It's not LARGER than every finite. >>>>>>>>>>> Which natural(s) is it "not larger" than", in the sense of not being a >>>>>>>>>>> proper superset of that natural or having that natural as a member? >>>>>>>>>> ....11111 binary (all bit positions finite) >>>>>>>>> Unless that string has only finitely many bit positions as well as only >>>>>>>>> finite bit positions, it is not a natural at all, as it is then neither >>>>>>>>> the first natural nor the successor of any natural, and every natural >>>>>>>>> has to be one or the other. >>>>>>>> It is the successor to ....11110. Duh. I've already proven that this is >>>>>>>> a finite value, given that all bit positions are finite, and that >>>>>>>> therefore no place in that string can achieve an infinite value, and >>>>>>>> that any such number has predecessor and successor. The cute thing is >>>>>>>> that the successor to ...1111 is 0, and that ...1111 is essentially -1. :) >>>>>>> Does it not bother you that nobody else agrees with, or even >>>>>>> understands, your proof? >>>>>>> >>>>>> I find it disappointing, but not surprising, that you don't understand >>>>>> such a simple proof, since it's contradictory to your education. I do >>>>>> find it annoying that you feel the right to disagree with it without >>>>>> understanding it. If you feel there is a problem with the proof, please >>>>>> state the logical error I made. If the string is all finite bits, and >>>>>> none of them ever can possibly achieve an infinite value, then how can >>>>>> the string have an infinite value? There's nowhere in the string where >>>>>> that can occur. It's that simple. Grok it. >>>>> 1) A finite string of 1s represents a (finite) natural number. >>>>> 2) An infinite string of 1s represents a (finite) natural number. >>>>> >>>>> 1) doesn't imply 2). >>>>> >>>> If the string is unbounded but finite, then 2) follows. >>> What's a finite but unbounded infinite string? I missed that you slipped "infinite" in here. Unbounded but finite may be considered potentially, but not actually, infinite. >>> >> One with all finite bit positions but no greatest. > > So... why does 2) follow from 1? Because then the string is considered finite. It's not finite AND infinite. > > 1) A finite string of 1s represents a finite natural number. For > example, 101 represents > > 1* (2^0) + 0* (2^1) + 1*(2^2) > = 1 + 0 + 4 > = 5 > > this representation bijects finite bit strings of 1s and 0s and natural > number. > > 2) doesn't work however... Take ...11111. Then the natural this > represents would be > > 1 + 2 + 4 + 8 + 16 + 32 + .... > > but there is no such natural. This sum is divergent. > The sum diverges to an infinite value as n APPROACHES oo. It is finite for all finite n. If all n are finite, it's finite. It's not infinite until n is infinite, since 2^n is finite for all finite n. Is n ever actually infinite? You sum diverges ****in the limit****.
From: Tony Orlow on 15 Sep 2006 15:58
Virgil wrote: > In article <45085df2(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Mike Kelly wrote: >>> Tony Orlow wrote: >>>> Mike Kelly wrote: > >>>>> It doesn't. I never said it does so please STOP asking that question. I >>>>> say that 111..... doesn't represent a (natural) value at all. >>>> If it is a whole number (no fractional component), is finite, and has >>>> successor and precedessor, then ....1111 is a finite natural. >>> It isn't. Now what? >> It's not what? A whole number? Show me the fractional component. Finite? >> SHow me the bit position where an infinite sum is possible. With >> successor and predecessor? There is no successor to this largest >> natural, unless the number line is taken to be circular. > > TO is claiming numerals (digit strings) are numbers, which shows that he > has no notion of what numbers are. > > And in order for a number to be a natural number, it must be a member of > the smallest inductive set, which means that it must be representable by > a digit string with both a first and last digit. > Proof: Let S be the set of naturals(i.e., the smallest inductive set) > representable by digit strings with a first and a last digit (which can > be the same digit). > (1) the first natural is a member of S > (2) if x is a member of S the so is the successor of x. > Therefore EVERY natural is representable by a digit string with both a > first and a last digit. Please give a concise definition of "number", so that I may mend my ways, O Virgil. You've never been able to before. There seems to general disagreement anyway. To equate the naturals with "the smallest inductive set" is artificial. The naturals are reals on the line identified by repeated offsets of the unit from the origin. They're real points like any others, identified with a successor function which is identified with unit offset along the line, and arithmetic increment. Without that notion of measure, you are not really describing the natural numbers as the real quantities they represent. |