An uncountable countable set
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From: Mike Kelly on 6 Sep 2006 04:45 Tony Orlow wrote: > Dik T. Winter wrote: > > In article <44ef3b88(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > > Dik T. Winter wrote: > > ... > > > > > > There is no such specific natural number. It is when we have them > > > > > > all, but as there is no largest number, this can not be achieved by > > > > > > taking them one by one. > > > > > > > > > > The set of all naturals numbers consists of only natural numbers. There > > > > > is NO natural number where the count becomes infinite. So there is no > > > > > point in the set, even if you COULD get to the "last" one, where any > > > > > infinite set has been achieved. > > > > > > > > And there is no point in the set where you have the complete set. Yes. > > > > Indeed. So what? > > > > > > So, if there is no point in the set which can even remotely be > > > considered infinitely far from the beginning, what makes it actually > > > infinite? > > > > What the set of all finite numbers makes infinite? The axiom of infinitys > > states that the set of all finite (natural) numbers does exist. From that > > is is easy to prove that that set is not finite, hence infinite. > > Yes, according to the Dedekind definition of an infinite set, but that > definition leads to some conclusions that offend some people's > sensibilities. > > > > > > If no element of the set can be an infinite number of steps > > > from the start, you may not be able to find an end. > > > > And indeed, when you go step by step you will not get at the end. > > Right, you will always have gone a finite number of steps and have > arrived at a finite natural. > > > > > > But does that mean > > > it's "greater than" every finite, or only "greater than or equal"? > > > > What is the difference? Assuming you mean "aleph-0" when you write "it", > > it is easily proven that: > > aleph-0 is greater than or equal to each natural > > gives the theorem: > > aleph-0 is greater than each natural. > > > > Because: suppose aleph-0 >= each n in N. Now suppose in addition that it > > is equal to some particular n. Well, n + 1 is in N, and so aleph-0 > > should also be larger or equal to n + 1. Hence it can not be equal to n. > > So it is not equal to any n at all. And so aleph-0 > each n in N. > > Only assuming that you have identified some largest natural. This is the > form of most proofs in this theory: assume a largest finite, find a > contradiction, and blame it something else. Of course you cannot find a > largest natural such that this is the size of the set. You also cannot > take an infinite number of increments without achieving an infinite value. > > > > > > > Indeed. The set of all natural numbers is just sufficient. > > > > > > No, it is far too great. If you have a countably infinite number of bit > > > positions, then you have an uncountably infinite set of strings. Where > > > bit positions are indexed by the naturals, the naturals are the power > > > set of the number of bit positions, > > > > Wrong. This is plain nonsense. Suppose there are three bit positions. > > The set of naturals {1, 2, 3} is sufficent to index them. In what way > > is {1, 2, 3} the power set of the number of bit positions (3)? > > Let me explain, using your example. We have the first three bit > positions 0, 1 and 2, representing 1, 2 and 4, the fist three powers of > 2. With those bit positions we can produce eight unique binary strings, > as follows: > > 0 1 2 > > 0 0 0 0 > 1 1 0 0 > 2 0 1 0 > 3 1 1 0 > 4 0 0 1 > 5 1 0 1 > 6 0 1 1 > 7 1 1 1 > > The set of binary naturals which can be represented by this set of bit > positions is the power set of the bit positions, each being a unique > combination, in the form of a sum, of those powers of 2. Given n bit > positions, we can produce 2^n unique strings. Given n unique powers of > 2, we can produce 2^n unique sums. Thus the set of binary strings is the > power set of the number of bit positions. > > > > > > No one is obligated to accept the theory at all. Whether it is proven > > > > to be "correct" or not, as I have no idea what "correct" in this context > > > > means. Is Euclidean geometry "correct"? Is hyperbolic geometry "correct"? > > > > Is elliptic geometry "correct"? > > > > > > Ah, now you bring up a prime example. Euclid set down laws for flat 2D > > > geometry, and questioning those axioms led to new shapes for space. > > > Accrdingly, the axioms of set theory might work together to describe a > > > system, but it is not impossible that entirely other systems might arise > > > from different starting assumptions. > > > > And indeed, I never did state the opposite. But if you want to get at a > > new system, provide axioms, definitions, and whatever. Tell us what > > axioms to retain and what axioms to reject. And if there are axioms to > > be rejected, come up with alternatives. > > Well, that's what I'm trying to do, to provide an alternative > perspective and system. I certainly reject the Dedekind definition of an > infinite set as being inconsistent with notions of infinite measure when > it comes to the finite naturals. By viewing the situation in terms of > density and range, the conclusion can be quite different. Since no two > naturals are infinitely distant, and since each natural occupies a unit > of measure on the real line, there cannot be an infinite number of them, > because you cannot fit an infinite number of unit segments in a finite > range. > > > > > > > But what is the case is that if you accept the axioms, you also have > > > > to accept what follows from the axioms. > > > > > > Yes, I understand that, and much to the consternation of some, I don't. > > > > Yes, much consternation, I can understand that. So apparently you are > > accepting the axioms, but not what follows from the axioms. What kind > > of logic are you using? > > I am NOT accepting the axioms. The axioms are artificial statements > which can be made to work together, but which do not follow from > elementary logic the way they should. Every axiom should be justifiable > based on fundamental concepts. I don't see that here. Which axiom of ZFC set theory do you find objectionable? Can you give an example of an axiom "justified by fundamental concepts
From: Mike Kelly on 6 Sep 2006 04:47 Tony Orlow wrote: > Dik T. Winter wrote: > > In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > > Dik T. Winter wrote: > > .... > > > > > Why do you need an axiom for that? Why is it > > > > > not derivable logically? > > > > > > > > Because without the axiom of infinity the set of naturals need not exist, > > > > and indeed, you can build a completely logical system with the negation > > > > of the axiom of infinity and with all other axioms remaining. It is > > > > similar to the parallel axiom in geometry. > > > > > > But without an axiom of infinity, it is demonstrable that, given the > > > axiom of internal infinity (continuity), x<z -> x<y<z, that any finite > > > interval includes an infinite number of points. Start with the line, and > > > identify points. There's infinity. > > > > 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 what does this mean? What is "the number line"? What is "left"? -- mike.
From: Mike Kelly on 6 Sep 2006 04:54 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> And yet, you are saying it's not technically correct. So, 1 is a natural > >> but not a rational or real? If rigorous formulations come to that > >> conclusion, then rigor does not ensure correctness. > > > > You're hopeless. You didn't understand a thing I said, which may be my > > fault for not providing adequate explanation in the context of your > > ignorance of the subject; but it is no my fault that you won't even > > look at a book on mathematics, such as even a introductory text in real > > analysis. > > > > MoeBlee > > > > Your whole point here is ludicrous. You are arguing that the naturals > are not a subset of the rationals, which are not a subset of the reals. > While each superset may require a more complex construction than the > subset, all elements of the subset are covered by the superconstruction. > As points on the real line, they are all real numbers. What you're > saying amounts to what I said above, which is indeed absurd. So, no, I > don't understand a thing you're saying, when you say naturals aren't > reals. Sorry. All natural numbers are finite ordinals. All real numbers are infinite sets of rationals. Obviously a set which contains only finite ordinals is not a subset of a set that contains only infinite sets of rationals. -- mike.
From: Mike Kelly on 6 Sep 2006 04:56 MoeBlee wrote: > Tony Orlow wrote: > > MoeBlee wrote: > > > Tony Orlow wrote: > > >> And yet, you are saying it's not technically correct. So, 1 is a natural > > >> but not a rational or real? If rigorous formulations come to that > > >> conclusion, then rigor does not ensure correctness. > > > > > > You're hopeless. You didn't understand a thing I said, which may be my > > > fault for not providing adequate explanation in the context of your > > > ignorance of the subject; but it is no my fault that you won't even > > > look at a book on mathematics, such as even a introductory text in real > > > analysis. > > > > > > MoeBlee > > > > > > > Your whole point here is ludicrous. > > No, my whole point just happens to conflict with the oversimplification > we were all taught in the fifth grade. > > > You are arguing that the naturals > > are not a subset of the rationals, which are not a subset of the reals. > > That is correct, given, as I said, that we're working with either the > Dedekind cut or equivalence class of Cauchy sequences method. > > A natural number is a finite ordinal. An integer is a certain kind of > equivalence class of naturals. A rational is certain kind of > equivalence class of integers. A Dedekind cut is a certain kind of > infinite proper subset of the set of rationals. An equivalence class of > Cauchy sequences is a certain kind of infinite set of infinite > sequences of rationals. Don't be abusrd. As any schoolchild knows, numbers are points on the number line. -- mike.
From: Mike Kelly on 6 Sep 2006 04:58
Tony Orlow wrote: > Dik T. Winter wrote: > > In article <44fdcaf1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > > Dik T. Winter wrote: > > > > In article <1157367096.604428.36330(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > > > A number has only one of the following properties: It is larger than or > > > > > smaller than or equal to any natural number. > > > > > > > > So omega and aleph-0 are numbers. They satisfy the definition. Thanks. > > > > > > So, they are numbers which are larger than any finite number? Why then > > > do we not consider an inductive proof of the form E y e N A x>y P(x) not > > > to prove P(aleph_0) or P(omega)? > > > > That would be a new axiom, and you may consider it, but it leads to > > contradictions when you retain the current definitions of aleph_0 and > > omega. > > Yes, I am well aware of that. That's why I have chosen to reject those > concepts in favor of finding something better, based on this > infinite-case induction. Noting personal. > > :) Better in what sense? What mathematics are you hoping to be able to do with your new foundation that cannot be done with ZFC? -- mike. |