I use the comma, because I am European. But if you prefer (like my computer and calculator) I can use the decimal point. let me get this straight... your last story leads me to conclude the same again: If the distance to travel equals 1 and any distance you have travelled is already too far away, thus you'll never be able to get from 1 to 0.999... (I think you're going to disagree) ERGO: 1 <> 0.999... Please Register or Log in to view the hidden image!
Whatever you prefer. Huh? How did you guess? Please Register or Log in to view the hidden image! Could you please reword that?
Alpha, my cousin is this leading anywhere? BTW what's whith the difference between calcuator and computer? I mean... calculating is not that different than computing, is it?
Merlijn: <i>Thanks James R, you just defeated yourself</i> Um, no. <i>Thus we can conclude: 0,999... < 1</i> No. 0.999... and 1 are two different notations for exactly the same number. There is no infinitessimal difference between 0.999... and 1. The ellipsis means that there is an infinity of 9s in 0.999..., and that makes all the difference. If the number of 9s was finite, I would agree with your argument, but it isn't. <i>And now for Phase Two. Why don't we differntiate the functions first, and then re-examine the problem?</i> Sorry, you've lost me again. Differentiate which functions? <i>PS... I was not very certain which notation to use.... Why are the math symbols not standard?</i> Which maths symbols? <i>SUM(i=0 -> infinity) [2^-i]=2 (which I thought to be closest to the original mathematical notation)</i> This isn't your problem at all. It's mine. What you wrote is clear enough. I would have written Sum (i=0 to infinity) 1/2<sup>i</sup> = 2 but its the same thing. As I said, I just mistook your i for the imaginary number rather than the index. My mistake.
similes and metaphors Alpha...as an example of calculating a never-ending number...possibly, never repeating number, is valid. If you prefer to disclude it, then focus on my comments that limitations on a formula change the original paradox. cheers, 137
Here are my thoughts on the subject 1<>0.99999... I am concvinced that 1 = 0.9999... from the definition of reals (please look at http://www.math.vanderbilt.edu/~schectex/courses/thereals) It seems that reals are defined as "Dedekind complete ordered fields". This definition uniquely defines reals up to isomorphism. Dedekind completeness is the principle that alowes us to take limits. There are several formulations for this property. One of which is that of Cauchy sequence. Every real can be denoted by a Cauchy sequence (this is a sequence of rational numbers for which the differences between the numbers become eventually arbitrarily small). 0.9999.. is a Cauchy-sequence as follows: 0, 0.9, 0.99, 0.999, 0.9999, 0.99999, .... The other number we are interested in is 1. An associated Cauchy sequence is: 1, 1, 1, 1, 1, 1, 1, .... Now, these two sequences are equal since the difference between the two can be made arbitrarily small. These two sequences are in the same equivalence class and are thus only different representations of one and the same number. Sorry. People that want 1<>0.9999 should consider another set built on top of the reals in which infinitesimals are contained. Again, look at the same website as mentioned before for details. In this set there are also the reciprocals of infinitesimals. Of couse in this set you loose the possibility to take limits (all limits?). Greetings, Han.
Still changing the parameters There are an infinite number of ways to set limits and define an infinite series into a manageable and solidified form. The mathematical formula of the endless halved distances still remains unfettered in the realm of pure mathematics. <1 will never equal 1 unless it is defined as equal to 1. When this type of defined limit is imposed, it only exists for the duration of its usefulness to the 'solver' of the inequality. Once the temporary solution is shelved, the secret world of the infinite smallness continues in its darkly mysterious and endless pathway. If you change the parameters of the orignally stated formula, you are not solving the unsolvable. If we set up a rule that says, without using calculus and defining limits, AND using a continuous halving formula of a distance A to B, will there ever be an un-rounded sum of all halves which equals the numerical distance between A & B? The answer based on these original paradox parameters must be no.
Re: Still changing the parameters That's the thing! As it's defined, it IS equal to one. The reason we are changing the parameters is because it was shown that there is no paradox as it was originally defined. We changed the original parameters ever so slightly to create a slightly altered paradox, which was also shown to be a fallacy. Because there can never be a true paradox! It only seems that way due to faulty logic and assumptions.
"That's the thing! As it's defined, it IS equal to one." We are powqerful being, but not that powerful.
infinities I have done a bit more reading on the subject of reals. There is a different presentation of "reals" (in fact it is an augmentation of the reals) including infinities. This follows better the intuition we have about reals and allows us to calculate with infinities and infinitesimals. This branch of mathematics was invented (discovered for Platonists) by Abraham Robinson in the sixties. Leibniz also calculated with infinitesimals, but never got it entirely right. Robinson did. Oh yes, the name for this analysis is "non-standard analysis (http://members.tripod.com/PhilipApps/nonstandard.html). There is also another kind of analysis which deals with limits in another way: Unified Analysis. I havent read much about it, but it seems very strange to me (for instance the limit of all the natural numbers = sqrt(2*pi) ). Greetings, Han.
What I meant of course was We are powerful beings, but not that powerful. With that I mean that I am a Platonists, meaning I believe that mathematical truths (or those in other sciences for that matter) are independant of the definitions we choose. It is possible we do not know them, and can only approximate them. I stronly believe that we do not just make up mathematics, but that we try to describe, to the best of our abilities, an objectvely true world. That's why Han wrote that a "branch of mathematics was invented (discovered for Platonists)". In fact, I do not see the use of doing sciences when you are not persuing objectively (not culturally, individually, or whatever) dependant truths! Han, you're a a smart and sensible person; how come you're not a Platonist.
You just gave the answer Please Register or Log in to view the hidden image! Seriously, the reason that I am not a Platonist is examplified by the discussion about 1<>0.99999... If you want the numbers to be equal then you choose to use the standard analysis, if you don't like this you just design another kind of analysis, as Robinson did. Greetings, Han.
but then Alpha, The fact that the mathematical setup is self-referenced to equal a totality does not erase the mathematical paradox. To slightly alter a paradox is still alterating a paradox. Can someone be slightly dead? Who ever said there can never be a true paradox? If you study your logic 101 then you will find plenty of logical paradoxes.... The mathematics of a line and numerical progressions are full of paradoxes. In Mathematics there are paradoxes...how can you deny this?
Because I've never seen a paradox in my life! Every paradox I've come across has either been a false paradox (a fallacy, it can be explained). There's always something wrong with the initial assumptions or something. A paradox only exists in abstract thought, not in reality. A paradox by definition can't exist. It just doesn't bloody well make sense! All those logical paradoxes can be explained.
my solution (sry I gave up reading further at a quarter of the discussion, maybe someone else said this already) I see it this way: By constantly dividing the remaining distance in 2, you also divide the remaining TIME in two. At the first step (halfway) you make, you set a maximum timespan: (time for step 1) times 2 minus (infinitely small amount of time). The reason why you will never end up where you want, is that the time it'll take to reach your destination (when using a constant speed, which you do as you divide time AND distance each step) will be equal to (time for first step) times 2. problem solved Please Register or Log in to view the hidden image! :bugeye: Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image!
All of you are assuming that you can divide distance into smaller fragments indefinately. What if everything is quantumized, including distance and time? Wouldn't this defeat Zeno's paradox since this math would not "fit" reality? Tom
