An infinite number of points on a line, an infinite distance, and infinite length of time are all undefined - the definition of infinite. Struggling to understand undefined is a senseless and endless endeavor. For instance, theoretically when the sun radiates energy one may consider the "rays" as emenating from the center of the sun which means the nearest neighbors of electromagneitc radiation from the sun do not arrive on the earth in parallel. So, what do we do when we calculate the angle between the raiating beams as measured on earth? We don't plug in the earth sun distance of 1.5 x 10^8 km and determine the radius of curvature all of which provides numbers for sober thought, but we simple assume the source is an infinite distance away, or that the radiation comes from a flat plane or whatever. We do this because it is impossible to measure ANY angle between radiation striking the earth. I know of no mesurements with detectors at the north and south poles that measure an angle between the radiation from the sun. Here is something to reflect on: If the earth were 1 cm in radius the sun would be 109 cm in diameter and located approximately 233 meters from the earth (2 1/2 football fields away). Mercury would be 87 meters from he tsun while Pluto would be approximately 9 km from the sun. The farther one travels from the sun does not alter the assumption that the sun is an infinite distance away as long as you are calculating the angle betyween neighboring lines of radiation, but you would use the correct distance to calculate how long it takes light to reach the earth from the sun. Undefined is not equivalent to unknown, as unknown can be real, but simply "unknown" like when you are playing blackjack and are deciding whether to take a hit or not, the next card on the deck is not undefined, because it is defined, it is a specific card though unknown as no one knows what it is until the card is ultimately turned face up. Or the exact chemical makeup of matter within a 500 meter radius of the center of the earth in unknown, and maybe even unknowable for the mere reason it is techologically impossible to achieve because it is too expensive to determine and no one is all that curious, but the answer is not undefined.
But, cannot one always add another slew of irrational to the current total and force the one providing he rationals to go round up more of her numbers? I think you should revise your statement to say that it is easier to provide rational numbers than irrational numbers and that there are an infinite supply of both.
The assumption was made in a general form, the actual list wasn't specified (just an example list was given). That means that for every list of real numbers we attempt to create, it will be incomplete. Therefore, it is impossible to create a one-to-one correspondence between the whole numbers and the real numbers. Since you can't make the correspondence, the sets can't be of equal cardinality. And since it was the set of reals that had members left over, it is the larger sized set.
Guys, Actually Cantor at his famous prove of uncountability of the set of rational numbers had used his method of diagonal arguments without any ties with any specific subset of those numbers (like it was done in GMontag's post). Cantor’ prove was based only upon one fact - the possibility of representation of any real into (0,1) as a finite or infinite decimal. His prove is much more beautiful than represented by GMontag. I told you: go and read "Physical Mathematics" on www.physics4u.com
There are indeed (from the abstract point of view) infinitely many levels of infinitely many collections of elements. But I wish to add that according to my point of view Alef0+1 > Alef0, 2^Alef0 < 3^Alef0, etc... To understand why I do not accept the Cantorian point of view about infinitely many elements, please look at: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf http://www.geocities.com/complementarytheory/Anyx.pdf http://www.geocities.com/complementarytheory/9999.pdf http://www.geocities.com/complementarytheory/Russell1.pdf http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
Yuriy, that paper is ridiculous drivel. The velocity addition formula has absolutely nothing to do with the topic. Addition of real numbers is a *defined* operation and is not constrained by physical laws. Also your second assumption is not an assumption at all, as it can be independantly proven.
Dear Gmontag, It is commonly acceptable that the enthusiasts of Physics could use language like you have used in your last post. What one can say: they are people of emotions. But such a language from someone who calls himself a mathematician (or an enthusiast of Math) is strange and unconventional. My article is analyzing Cantor's prove, shows that there was done two, not the one assumptions, shows that Cantor had missed the second assumption he actually used and consequently had came to logically controversial conclusion. That there is a possibility of another structure of Math that is adequate to Nature. And that the Einstein's law of addition of velocities is the first sign of Nature what Math it is based upon... All what you can (and must do) as a professional – reveal a proof that my analysis is wrong. A simple accurate proof, not a philology of a proof… So, please, be professional and good luck, my friend…
The assumption you claim Cantor made: "B) the interval (0,1) contains any real number, which can be expressed as a decimal with any, finite or infinite, set of digits after a decimal point" doesn't make any sense. to me. What are you trying to say here? That there are some infinite decimals 0.x<sub>1</sub>x<sub>2</sub>... that are somehow not on the interval (0,1)? That's the least ludicrous parsing I can manage, but it would mean you don't understand the completeness axiom of the reals (or maybe what a decimal expansion actually is).
Dear shmoe, Please, at first read the mentioned article again. It will prevent you asking a lot of inappropriate questions... I’m not trying to say, I am asserting that namely this conclusion comes out from the Cantor’s experiment as a possible discovery. ( And “the completeness of axioms of Real numbers and what a decimal expansion actually is” is just what we should discuss under knowledge of a new possible interpretation of the results of Cantor’s experiment. The article is just about that, no matter how bad you fill about it). All, what one should do to disprove propositions of my article, is to prove that these two propositions lead to some wrong description of any fact of Nature. Can you do that?
Right, I didn't read it. I magically managed to quote you though. I'll ask again, do you believe that there are some infinite decimals 0.x1x2... that are somehow not on the interval (0,1) (I should say apart from 0 and 1)? Is that what you were trying to say was wrong with "B"?
Sorry, you have answered faster, than I could edit my post. So, as I told people in my article, Cantor might discovered the fact that A. Set of Real numbers is countable, and B. There are “holes” between Real numbers in sense of orthodox Math, when not all possible decimal exists. (His number J, which was constructed in his experiment was a first example of such decimal) Two propositions I did in the end of article are devoted to show how and why it happens.
There are no "holes". This is independant of whether the reals are countable or uncountable. This will follow from the ground up, loosely speaking- build Naturals out of sets, Integers from the Naturals, rationals from the integers and finally the reals from the rationals. If you've ever looked at the details it's clear that the real numbers are a complete metric space. This is done before you attempt to prove the reals are uncountable, otherwise you don't even have reals to talk about. Any decimal expansion you like represents the limit of a Cauchy sequence and therefore has a unique real number associated with it (though a real number may have more than one decimal expansion-it's "tail" will be all 0's or 9's), Cantor's J is such a decimal. Therefore it's a real number. As long as you don't pick all the digits to be 0 or 9 (and we can easily do this, just make all digits 4 or 5) you're in good shape. So we can rule out your B possiblity.
Dear shmoe, It seems to me that you really do not understand the problem we have touched. Listen to me carefully: TRY TO PROVE ANY OF YOUR ASSERTION BEING BASED UPON THE RULE OF ADDITION THAT I HAVE PROPOSED IN MY ARTICLE. You will immediately see what a problem we deal with...
Yuriy, as I already told you, addition is an operation that is *defined* for the reals. It's part of the reals definition as a field. Its not subject to change. You are of course free to make up a new type of mathematical object that follows what ever rules of addition and multiplication you wish, but you wouldn't be talking about real numbers then, and you wouldn't be talking about Cantor's diagonal proof either. P.S. Your proposition that numbers follow the velocity addition formula also belies a fundamental misunderstanding of the velocity addition formula and where it comes from. Velocities aren't added that way because of some magical new addition formula. They are added that way due to frame changing and the Lorentz transformations. If you are not changing frames, you still add velocities the old fashioned way.
Dear GMontag, 1. I started my article with citation of V. I. Arnold, one of the most brilliant and respectable mathematicians alive (do you recall: “Math is a part of physics. Physics is an experimental science…” and so on). Do you know why? To emphasize that I am talking about Mathematics that pretends to be a quantitative reflection of things and their relations in Nature. If you are talking about some axiomatic Math, which has nothing to do with reality of our World and reflects some logical (or intellectual, if you want) abilities of human brain, you can establish it upon any set of axioms. But we (at least, I am) are talking about Math that Arnold was talking about. 2. You said: “addition is an operation that is *defined* for the reals”. Yes, and I propose to redefine it to match our Nature. You said: “Its not subject to change”. Why not? What will be wrong if we will do that? Will Math, based upon new definition of operation of addition, describe Nature worse than old one? Or may be better? That is a central point of my article. It will destroy some cozy theorems, definitions and so on of old Math. So, what? Who cares, if we will get the better quantitative description of reality. BTW, we knew real number far before we invented such notion as “numerical field”… Who cares if reals will not form field in respect to new operation of addition? 3. The second clause in your post I do not understand at all. Why “wouldn’t be” if it is? Because you do not like it? Or what connection between Cantor’s brilliant method of diagonal arguments and definition of operation of addition you see? 4. I do not touch your P.S., because it is matter of a pure physics and by my professional opinion namely you have shown there your fundamental misunderstanding of the velocity addition formula. 5. But the major impression all that posts gave me is that it is to complicate to speak on the level we started. Let start from much more easy one. My first proposition, particularly, leads to the following conclusion: there is a finite number of the natural numbers. Can you disprove this assertion?
First of all, I don't care what anyone says, math is *not* a part of physics. Physics studies the natural world, math studies abstract concepts and the relations and interactions between them. Just because math is very often used as a tool in the physics does not mean it is a part of physics. Second, *all* math is axiomatic. Math is the process of proving theorems, and that is impossible without axioms. What will be wrong is that you will no longer be talking about real numbers. The term "real number" has a very specific definition. It is not something you can just redefine without speaking gibberish. Like I said, you are free to come up with a new mathematical object that has your new definition of the addition operation, just don't call that object a real number, because that term is already taken. What are you talking about? The definition for the addition operation on the reals is not some math-wide definition. It applies only to the reals. Other mathematical objects have their own definition of the addition operation. Yes, there may be some physical application of your new mathematical object, but that doesn't change that fact that it is *not* a real number. The problem is that Cantor is talking specificly about real numbers and you are not. As I have said before, real numbers have a very specific definition. If you choose to use some other definition, you are no longer talking about real numbers. Relativity uses the same addition operation as the real numbers do. The velocity addition formula is *only* for changing frames. Certainly, but I'm holding you to the mathematical definition of a natural number: 1. Assume there are a finite number of natural numbers 2. Then there must be a greatest natural number, x. 3. Let y = x+1. 4. y>x. 5. 4 contradicts 2, therefore by reductio ad absurdum, premise 1 must be false. QED
You claimed Cantor was ignoring the possibilty that the reals have "holes". They don't. It's that simple. You can make whatever claims you want about your attempted new addition, but realize that you are no longer talking about the reals with this operation(see GMontag's posts).
GMontag, 1. “First of all, I don't care what anyone says”. So, are you inviting me to speak on the same language? Please, get the response: I don’t care what you think about origins of mathematics. 2. “Just because math is very often used as a tool in the physics does not mean it is a part of physics.” Of course; but Arnold was spoken about historical and natural origins of Math, not about opinions of nova-axiomatics. 3. “Second, *all* math is axiomatic. Math is the process of proving theorems, and that is impossible without axioms.” Professionals have another opinion (read Arnold and many others his colleagues). But it is not important now – read clause 1 here. 4. “What will be wrong is that you will no longer be talking about real numbers. The term "real number" has a very specific definition.” According my knowledge, the classics, who discovered and studied a lot features of real numbers long before Cantor, never were so categorical like you are. But it is not important now – read clause 1 here. What is important – please, be more specific and give us details what you mean under these categorical assertions in form of proof as it is use to be acceptable in Math. 5. “What are you talking about?” Are you kidding me? So, you do not understand real numbers anyhow else as through numerical field? Well, let me remind you that definition of numerical field and prove that real numbers form the field was done by Georg Cantor in 1877. Let us assume we are leaving in 1850, i.e. we already knew Math as it was accomplished due to Gauss, Jacobi, Dirichlet, Cauchy, Riemann and many other genius predecessors. Would I be allowed to set up the same questions, or not? I guess I could. Would you answer me as you do now? Of course, you will not. Simply because in that times people were able to do a lot with real numbers do not knowing the nova-days axiomatic “occupation” we have now. How you would answer me then? If you do not know answer – say it strictly, do not play philology with me and others…. 6. “If you choose to use some other definition, you are no longer talking about real numbers.” No, my friend, I will speak about the same real numbers, but I will use the definition of them, which was in use by all grate mathematicians before 1877: “the real numbers are those numbers that are the coordinates of points on the coordinate axis”. And this definition comes from Physics, part of which the Mathematics always was and always will remain. Why this definition is better then any other? Because: a. it is operational; b. it directly selects only such Math, which describes a real World: is physical point on the axis - is real number, no physical point on the axis – no real number; c. it allows to ask the Nature: what real numbers you know (have, allow as measure of your features, etc.), i.e. it allows to check Math experimentally. And it contains any feature of real numbers that contains any nova-days axiomatic definition. 7. “The velocity addition formula is *only* for changing frames.” I already told you my opinion on that your understanding of velocity addition formula. However now I have to remind what I said in my article: lets measure velocities in gauge of c. Then Einstein’s addition formula will be rule of addition of some specially selected (from point of view of Physics!) fractions. It gave me the idea, nothing more. It allowed me to formulate my two propositions, nothing more. It did its job. Let us do oures… 8. And finally, your disprove of assertion “The number of the natural numbers is finite” You have repeated famous Pythagoras proof: “a. Assume there are a finite number of natural numbers b. Then there must be a greatest natural number, x. c. Let y = x+1. d. y>x. e. d contradicts b, therefore by reductio ad absurdum, premise a must be false. QED” And now add my second proposition about new type of addition of real numbers. Whole your construction starting d. becomes invalid. Therefore, you can not yet disprove these two propositions acting jointly. And my guess is, you will not be able to do it anyway. And that was the essence of whole article… The main question remains the same: which Math – conventional or based upon Einstein’s addition – better describes the Nature?
