We've been giving you explanations. I had thought you were simply not understanding them, but apparently you're just ignoring them. Infinity is treated in a manner consistent with the rest of the system of mathematics. If you're not willing to accept that system, then feel free to define your own (though it seems from other threads that you may have tried to already). If you're going to argue within (not with) that system of mathematics, you need to accept and understand the rules of that system. So far you've shown that neither is true. And if you would instead argue with the system, you need to specify your own rules and explanations for your definition of infinity.
1100f: Take it up with Wolfram, then Please Register or Log in to view the hidden image! http://mathworld.wolfram.com/Undefined.html Also, this says that 0*0, 0*i, and i/i are indeterminate: http://mathworld.wolfram.com/Indeterminate.html
Infinity doesn't exist If then i+f=i+i i=f Or i is finite or f is infinite. Since we have only examples of the finite in the real world let's conclude that infinity does not exist.
I agree with you. Except that 0*0 is perfectly defined, since 0*0 = 0. I guess that it was a a typo mistake and that you meant 0/0
1100f, quote: -------------------------------------------------------------------------------- Originally posted by malkiri I may be mistaken, but I believe f / 0 is undefined, rather than infinity. -------------------------------------------------------------------------------- you are mistaken, f/0 is infinite. 0/0, i/i and 0*i are undefined You have hit my question on the head here. f/0 = i is a valid process but the algebraic inversion f = 0 x i is indeterminate. The rules of infinity are not reversable. that is a = b but b does not = a. If we say 7 x 8 = 56 then 56/8 = 7 but in the case of infinity that process fails. Knowing to believe only half of what you hear is a sign of intelligence. Knowing which half to believe will make you a genius.
Re: Infinity doesn't exist you cannot just cancel the two i's on both sides of the first equation as you do it for real numbers since for the cancelling, in fact, you substracted i from both sides. So that what you have is: i + f = i + i ==> i + f - i = i + i - i ==> undeterminate = undeterminate (and not i = f), which is ok.
In moving from f/0 = i to f = 0*i, you would need to multiply both sides by 0. If we were talking about numbers other than 0, we would cancel the fraction on the left and leave f. But 0/0 is not 1, so we wouldn't get f = 0*i out of that. 1100f: Yeah, I meant 0/0. Please Register or Log in to view the hidden image!
Hello Crisp OK then 0.5 is a real number. Thought you were angling at digits. Personally don’t think of half as a number. What you are getting at has been covered in this thread. Infinity, if it exists, lies in between 0 and 1. It is the sum of a never ending infinitesimal approach to zero. Allowing, that is, that you are suggesting that there is an infinite number of intervals between 0 and 1 and twice as many between 0 and 2. Do you understand that your whole logic is being built on zero being a real number Crisp. If you disallow yourself from 0 being comparable to 1, your mathematical logic becomes the nonsense that it is. Infinity: An unbounded quantity greater than every real number... The problem of contradiction that you face is you want more than one of these of these as per given in Phoenix 2634’s definition. Address that and the fact that you are using zero as a real number and you may be able to start discussing whether or not infinity actually exists. Apologies for not using the same relay station as you with respect of digits and numbers. No point entering semantics over that. Regards Leeaus.
Hello Malikiri You suggest that the rules of a system can not be questioned from within the system. Mathematics without freedom of thought would just be a system of rote. Infinity: An unbounded quantity greater than every real number... If that is the definition of infinity that the system of mathematics is using, all that is being asked is how can than be more than one of these unbounded greater quantities. No explanation of answer to this question has been ignored. The contradiction of terms when several infinities are considered is apparent. Just a question from the back of the class room if you like. Regards leeaus
"OK then 0.5 is a real number. " Ok, one step closer to seeing the light. "What you are getting at has been covered in this thread. Infinity, if it exists, lies in between 0 and 1. It is the sum of a never ending infinitesimal approach to zero. " I do not have to say I completely disagree. And I am not getting to your result. The point I was trying to make is the following: if you accept that there are an infinite number of real numbers in [0,1] and there are an infinite number of numbers in [0,2], which has twice the number of real numbers as [0,1], then you are forced to conclude 2*infinity = infinity. Now, how many real numbers are there in the interval [1,2] ? Now you'll see that your reasoning of "infinity is between zero and one" does not hold. You can write [1,2] = [0,1] + 1, hence you get that the number of real numbers in [1,2] is the same as the number of real numbers in [0,1]... being infinity. "Allowing, that is, that you are suggesting that there is an infinite number of intervals between 0 and 1 and twice as many between 0 and 2." I am not only suggesting that, you can prove it easily by the method above: denoting "#" as "the number of numbers in..." (mathematicians refer to this as the cardinality): #[0,2] = #[0,1] + #[1,2] (disjunct sets) = #[0,1] + #([0,1] + 1) = #[0,1] + #[0,1] (one-to-one maps preserve cardinality) = 2*#[0,1] It is not 100% accurate (you should leave out 1 in the interval [1,2] , but this is a correction of 1 to infinity). "Do you understand that your whole logic is being built on zero being a real number Crisp. If you disallow yourself from 0 being comparable to 1, your mathematical logic becomes the nonsense that it is." I do not think anybody has ever claimed that 0 is not a real number. In mathematical terms: the real numbers are a group with respect to addition, so there is a neutral element in that group which we denote "0". It is not only logical but also needed for consistency that 0 is a real number. "The problem of contradiction that you face is you want more than one of these .... What problem ? What contradiction ? ... Address that and the fact that you are using zero as a real number and you may be able to start discussing whether or not infinity actually exists." Infinity does not exist as a real physical quantity. Whoever told you so was wrong. I am confident that nobody ever claimed that in this thread anyway. Infinity is a mathematical abstraction. You are the one trying to attribute something "touchable" or physical with it. Bye! Crisp
In fact 0.5 is our decimal way to represent the number 1/2, which is a rational number (and of course a real number, since the set of rational numbers is a subset of the real numbers. This is your personnal view. However, mathematics is based on axioms and definitions. All the mathematics based on the definition of natural numbers by Peano axioms, their generalization to integers and then to rationals and then to real numbers (using the Dedekind cuts) is perfectly well self-consistent. What you are trying to do is to show flaws in the mathematics according to some other axioms and definitions that you invented. Your definitions, for example, of infinity is not consistent with the usual definition of infinity. the number 3/4 lies between 0 and 1, while the sum 3/4 + 3/4 does not. So infinity does not lie between 0 and 1. You have a definition of convergence of series according to this definition the sum of the harmonic series does not converge, and by definition "converge" to infinity. The "number" of real numbers in the segment [0,2] (the cardinal) is infinite. In the segment [0,1] there is the same number of elements since there is a 1-1 maping from the segment [0,1] and [0,2]. But when you take the segment [0,1] it has half the number of elements than in the segment [0,2]. So obviously, half infinity is infinity. By definition 0 is a number (in fact if you look at Peanos axiom, the first of them says that 0 is a number) As I told, in all this thread, you are trying to find flaws in tmathematics based on some axioms and definitions by using a different set of axioms and definitions. BTW your definitions are even not conistent since you defined infinite length as:
Hello 1100f and Crisp Surprised that you readily accept zero as a number. It is the absence of a number. When the absence of a number is allowed to be a number for the sake of convenience, something of no value is given value. 0, 1 and infinity are all there is to things. The trick is to reach an understanding of how each relates to the other. Peano’s axioms fail at this level with an equivalence between 1 and 0. Leeaus will die happy with his understanding of the relationship between 0, 1 and infinity. 1 is the real number, zero is the absence of the real number and infinity is the link between the absence of the real number and the real number. Leeaus axioms, which are not exactly irrelevant to the non existence of infinite distance. A/ 1 is the only real number B/ 0 is the absence of the real number C/ Infinity is the numerical connection between the real number and the absence of the real number. D/ 2,3,4…. are a sub set of the real number. They are within 1, not beyond 1. 1100f, glad that you give direction back to the original question, that of the existence or otherwise of infinite length. To clarify, each of the three dimensions would need to encompass all distance if they were of infinite length. What you refer to is a comment about the JR mish mash between a line, an x axis, and a length L. Call your infinite length what you wish, a line, a length or an axis. Just make sure it has both ends open so as it can encompass all distance. Regards leeaus
When using numbers to count objects, zero is the absence of an object. It is not an absence of a number. It is still a number. no absence of a number is allowed to be a number There is no equivalence between 0 and 1. However there is a relation between them as according to Peano's axioms as zero is the predecessor of one. I wish you a very long life, so that you migth study mathematics in order to understand it. Obviously you are not talking about real numbers, so please don't say 1 is the real number, zero is ... Your axioms are really obscure. You do not say in your axioms what they define. Do they define a n object, a set of object? I don't understand how a set {2,3,4...} can be a subset of a set containing one element. The more I read your axioms, the less I understand them. I really don't understanf what is "a numerical connection between the real number and the absence of the real number". Please don't call them the real numbers, since you cannot use something that is well defined and replace it by something that has another definition. Complex numbers are well defined, Quaternions are well defined, Boolean numbers are well defined, Real numbers are well defined, Rational numbers are well defined, Integer number are well defined, natural numbers are well defined. Your axioms are not equivalent to any of these numbers, so please don't call the number in your definition/axiom real number. Sorry, but when you say that I may call my infinite length what I wish, it is really not a serious argument because a line, a length or an axis are three different things. If you want to talk about mathematics and you don't understand that mathematics is something rigouros, then, don't be surprised if people will talk about crackpots. And if you give some definition of something, don't change it every time people show you that there are flaws in your definition. (definition 1: A line of infinite length would encompass all distance, where length has the meaning of dimension length, width and breadth. definition 2: A line of infinite length would encompass all distance, with length as a distance. definition 3: A line of infinite length would encompass all distance, just make sure it has both ends open so as it can encompass all distance)
Hello 1100f Easiest and probably best to pronounce self a crackpot and get the hell out. Your post is not particularly coherent. Zero is a number that represents the absence of a number is about the best we can do with your insistence on zero being a number. Peano’s axioms fail at this level with an equivalence between 1 and 0. If zero is a number and 1 is a number according to Peanos, then the equivalence is said according to Peanos. What you say different? 1 is the number if you like. Real is perhaps superfluous if that troubles you. The axioms define the reason of the universe. Sorry that you find them obscure. I don't understand how a set {2,3,4...} can be a subset of a set containing one element. Not sure how to take you there. Imagine a pixel on a computer screen. That is one element. The dots that make up that pixel are the sub set. The analogy is not exact but it may help. Or do it with stars. 1 universe, the stars are the sub set. Semantic arguments about what is meant by length are a bit tiresome. Hold your hands a part and you may begin to get the idea. Call what is between your hands length or distance or an x axis. Doesn’t matter. The question is whether or not what separates your hands could be infinite. Your hands are not part of the question. Just trying to convey what is meant. Making out differences between length, distance or an axis is academic. Whether space is infinite or finite is where the real question lies. You should try harder with the axioms. BUt if it makes 1100f and others happier in their private worlds, leeaus is a crazy crackpot. Space is infinite. Such has been proven extraordinarily well by all and sundry on this thread. Regards leeaus
leeaus, You are making the same mistake as many people who have come before you on this board claiming they reinvented the hot water: they simply don't have a clue what they are talking about. I'm sorry, but you cannot do a simple thing with the axioms you have provided. I don't see a possible way to concatenate them to form a proof of a basic premise as 0*x = 0. Once again, you are trying too hard to make it all "physical", which your "zero as the absence of a number" suggests. Also a flaw of many people who have come before you. I'm fine with the number systems I use, so if you want to discuss any further on the nature of infinity, I suggest you first learn how commonly accepted number theories work. If you have done that and you disagree, you'll find it more easy to find the flaws there (if any) and construct your new theory if that is really what you want to do. Bye! Crisp
Hello Crisp The axioms stand. Your infinity ones don’t. If you can’t understand that 1 is the only number yet, it is unlikely that this thread will be your enabler. You haven’t even proved that 1 + 1 = 2 Put some work into your personality and your spelling of concentrate. Reinvent hot water is not analogous with what was being done. A suspected proof of infinite distance was supplied. The supplier was not prepared to answer questions about it. That was not analogous with a reinvention of hot water from this side. The bottom line is you don’t know the reason of the universe but leeaus would be able to explain that finite space is the reason. That is where things end. Regards leeaus
