An uncountable countable set
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From: mueckenh on 28 Sep 2006 06:04 Dik T. Winter schrieb: > > > > > Educate me. What are they? Pray provide sources. But your rubbish > > > > > is not my rubbish... > > > > > > > > Cantor's truths are self-evident truths which cannot be changed > > > > arbitrarily in contrast to the axioms of modern set theory. > > > > For instance: I + I = II. > > In some cases, yes. In other cases, no. Depends entirely on how you > define "+" and the other symbols. In Greek mathematics I would expect > I + I = K (as in 10 + 10 = 20). > > > > > > > Yeah, whatever. Is this a reply to my question? > > > > That was my intention, yes. If I failed, you should improve the > > precision of your question. > > I ask for self-evident truths. Upto now you have not provided any. For Cantor I + I = II is such a self-evident truth. (Of course with the usual meaning of "+" and "=".) For me too. Or take another one: If you divide a sphere in a few parts and afterwards put them together again, then you will get one sphere and not two. But if you dislike these examples and do not agree, then take your reaction: It is a self-evident truth that you do not agree. Regards, WM
From: mueckenh on 28 Sep 2006 06:06 cbrown(a)cbrownsystems.com schrieb: > Tony Orlow wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Mike Kelly schrieb: > > > > > >> It is not a proof. Division is not defined where either operand is an > > >> infinite cardinal number. > > > > > > But you can conclude n / aleph_0 < 1 by inserting aleph_0 > n which is > > > definied *if aleph_0 is a number in trichotomy with natural numbers*. > > > > > > You cannot have both, assert that aleph_0 is a number larger than any n > > > but on the other hand prohibit that the inequality n < aleph_0 be > > > utilized. > > > > > > Regards, WM > > > > > > > I agree. If x is infinite, and that means greater than any finite, and > > trichotomy holds, then 1/x is in [0,1], and is real, though > > infinitesimal, of course. But, I am getting the feeling that set > > theorists now don't want to claim that infinite is greater than finite. > > I think it would be more correct to say that set theorists don't now > want, and have never wanted, to claim that some mathematical object is > greater than another mathematical object without an /explicit > definition/ of what is meant by "greater than" in the context of the > discussion. There is only one common meaning of "greater than" in arithmetic: If you subtract a from b, and the remaining is positive, then b > a. In set theory we say set a bijects with a subset of b. In the same fashion you can proceed, saying b/a > 1. There is no special definition of division required to determine whether a ratio of trichotomic numbers is larger or less than 1. Regards, WM
From: mueckenh on 28 Sep 2006 06:08 Dik T. Winter schrieb: > > The successor function *is* counting (+1). > > Wrong. After a while you will have run out of the predefined successor, unavoidably. Then you have no other choice but to add 1 each time you proceed. That is counting. > > > The successors are defined > > without counting only over a very restricted domain. In the usual > > decimal systems only from 1 to 12 and then repeating again and again > > from X to X + 9. > > You think so because you again focus on the decimal system. I wonder how > you get at 12. German influence? Of course, but not only in German or English we see that phenomenon. I would be surprised if it were different in Dutch. Regards, WM
From: imaginatorium on 28 Sep 2006 06:08 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Consider a (notional, theoretical, mathematical, not physical) x-y > > plane. That is, an area in which there is a point (0,0) in some > > particular place, an x-axis, y-axis, and points are identified by > > coordinates x and y, using (in normal maths) real values for these > > coordinates. Consider (for convenience) that this plane is embedded in > > a notional graphics application, with a "Fill" function. So if we draw > > the circle x^2 + y^2 = 49 (centre origin, (constant! Zick, be quiet!) > > radius 7), then click with the Fill function on the point (2,1), it > > fills the circle, and no paint spills outside that radius 7. > > > > Now suppose we have the graphs of x=2 and x=5. Vertical lines, > > extending up and down without limit. Suppose we click with the Fill > > function on the point (3, 4), what would you say happens? Obviously > > paint fills the vertical strip of width 3. Would you say that any paint > > was able to "spill" around the (nonexistent!) "top" of either of the > > graphs, and somehow fill more of the plane than this strip, or would > > you say we just get a (vertically) unbounded strip of blue? (Goddabe > > blue!) > I'd have to agree that it would fill the strip only. Proceed, but it > would be nice to know the context of the question. Ok, well just as a diversion: suppose you were on sci.comp.graphics.crank, and one of the residents produced a long, rambling argument, including mention of Planck's constant, twin-slit experiments and more, at the end of which was a claim that outside the strip would also be a very pale (ok "infinitesimally pale"!?) blue. How would you try to justify your claim that the blue fills the vertically unbounded strip only? Note that when discussing the behaviour of a real-world graphics program, within a bounded window, it's possible to discuss the paint-filling as a terminating procedure. With an unbounded strip, it obviously isn't. So I would say something like the following: for the paint to spill outside the vertical lines bounding the strip, there must be a path from a point inside to a point outside. But since the x-coordinate of the points on the path must go from (say) 4 to 6, at some point it must be 5; and that point must be a point on the boundary, so it would have crossed the boundary, and it's not allowed to cross the boundary, so this can't have happened. ---- back to the point ---- Now consider some other graphs: y=1/x, fill from the point (0, 0) - get blue lower left and upper right quadrants, plus filling out to the white lobes that almost fill the upper left and lower right quadrants. OK? (Graph is a hyperbola) Now consider the following two hyperbola-halves: y1 = -1/x (for negative x) y2 = -2/x (for negative x) Each of these is a lobe in the upper left quadrant, OK? Clicking on (-23, 34) would fill just the lobe formed by y2=-2/x (since this curve is always above and to the left of the other one); clicking in the lower right quadrant would fill three quadrants, and the area up to the y1 curve, leaving a (slightly larger) upper left white lobe. (I hope all this terminology is clear.) Would you agree that clicking on (-1, 1.5) fills the sliver between the two hyperbola lobes? (I say "sliver", though the area is infinite, since sum(1/n) doesn't converge - plus a bit of hand-waving.) Do you see a connection to the original problem? Brian Chandler http://imaginatorium.org
From: mueckenh on 28 Sep 2006 06:12
Dik T. Winter schrieb: > In article <1159186907.615747.304410(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > > > > Let me ask you to, before I can answer such a question. What > > > > > > > is your definition of "number"? (I asked the same from > > > > > > > Wolfgang Mueckenheim, but his answer was not satisfactory, > > > > > > > also not to himself, I think, because he never answered to > > > > > > > questions about it.) > > > > > > > > > > > > Didn't you read my paper on the physical constraints of numbers? > > > > > > > > > > Yes, I read it. Mathematically it makes no sense. > > > > > > > > You did not yet recognize it, perhaps later. But you were not telling > > > > the truth above, were you? > > > > > > Where did I not write the truth? > > > > Here: > (See above) > > > > If I considered Dik as one person, I would write {Dik, Virgil, me} for > > > > instance. But you are right, there is some ambiguity. > > > > > > Yes, so it is not a proper definition. And as such, the above to me still > > > makes no sense as a proper definition at all. > > > > There is a natural number which is the largest one ever mentioned or > > thought during the lifetime o the universe. It is not properly defined > > before the universe ceases and probably also not afterwards. But it is > > or will be (depending on the question of determination or not). > > Nevertheless, it does not exist yet. We have to live with those > > improper objects. > > So, again, no definition. Where did I not speak the truth? Here: "...because he never answered to questions about it". Most questions on the representation of a number are answered in my paper. But I am always ready to respond to further questions. If my definitions are not proper enough, according to your taste, then reality is to blame for that. anyhow, your assertion "he never answered to questions about it" is a lie. > > > > > > > 2) or is completely determined by a series of digits. > > > > > > > > > > Question. A terminating or a non-terminating series? > > > > > > > > There are only terminating series. There is no infinity in reality and > > > > useful mathematics. > > > > > > Oh. So you state. But 1/3 is a number? > > > > 1/3 is a number, properly defined, for instance, by the pair of numbers > > 1,3 or 2,6 or 3,9 etc. But 0.333... is not properly defined because you > > cannot index all positions, > > Again, you *ignore* the definition of that notation as a decimal number. > I state again, that notation has *no* meaning until some meaning has been > defined. In mathematics it is defined as the limit of a sequence. If > you think that definition is invalid, you should seriously consider all > use of limits in mathematics to be invalid. The definition of an object does not provide its existence. The set of all sets is well defined. Nevertheless, it does not exist. > > > > What do you mean with "exisiting"? > > > > Existing is a thing you can use, like the largest known Fermat-prime, > > or the theorem of Pythagoras, or a hot dog in your hands. > > Eh? > > > > The set of prime numbers is infinite > > > and unbounded. The set of known prime numbers is finite and bounded. > > > > it is finite, but not bound, because it can, and probably will, grow. > > That is the same as with the set of natural or real numbers. > > The set of known prime numbers is bounded. Period. It is a specific set > that now and today consists of a fixed number of elements. It may be a > different set tomorrow, but that is something different, and again, > tomorrow it will be fixed and bounded. The cardinality of the set of prime numbers known today P(t) is as unbounded as the time variable t of today. Of course it has a present value, but that is changing. Of course you can say time tomorrow is completely different from time today. But that does not make time different. If we find another prime tomorrow, you can be sure that those primes known today will remain exactly the same. No change. That's all. Regards, WM |