Libertarian
Your error in the first proof is in
It should read, instead:
Premise 2: 10X = 9.9[n-1] (by Axiom 3)
Once you make this correction, both proofs come to the same conclusion. Unfortunately, in neither one do you prove anything about a decimal point followed by an infinite number of 9’s.
Hi Spiritus and Gilligan, keep up the good work.
.
Libertarian:
As Dark Wing Duck mentioned, I didn’t mean to imply there is a 1-1 correspondence between the reals and the integers (there isn’t). Both of these sets, however, do have the following property–Each has a proper subset that can be put into a 1-1 correspondence with the original set. The integers can be put into a 1-1 correspondence with the positive integers (a proper subset); the reals can be put into a 1-1 correspondence with the interval (0,1), for example. The integers are a proper subset of the reals, but cannot be put into a 1-1 correspondence with the reals (not just any proper subset will work, in other words).
This property captures completely the notion of an infinite set.
4 times .7… = 3.1…, but I don’t see the point.
Lib, where exactly did you get your belief that there are groups on “infinists” and “anti-infinists” doing battle?
What I find extremely funny is that fact that you first say that Infinity is not a number but a concept, then you try to use an(extremely faulty) pair of matematical equations to prove your point.
This just in: Dark Wing Duck, building on the groundbreaking work of Libertarian from early yesterday afternoon, has just discovered the smallest positive number. Let’s hear it in his own words:
“This result would not have been possible without the sustained effort of a large group of researchers, and I wish I had time to thank them all personally. What we have found, in short, is a new smallest positive number, ABS(0.0[n+2]1). I’d like to point out that this is only one-tenth the size of the previous smallest known positive number, and that one was a really weeny little thing to start with. This discovery should have enormous repercussions for science, industry, the whole global economy in fact. I think it’s safe to say we’re entering a new age.”
Q: Dark Wing, is this the end? Will a smaller positive number ever be found?
“We learn never to say never in science. One has only to gaze at the vast, though finite, abysses of space, to consider the sweeping, though finite, panorama of time itself, to know that there may be one or two things we haven’t discovered yet, although we’re obviously getting down to the butt end of stuff we can find out.”
Thank you, Dark Wing Duck.
…
(hey! ABS(0.0[n+3]1)!)
Because whatever n is, ABS(0.0[n]1) is less than ABS((0.0[n-1]1)). Hey, I just discovered the smallest positive number!
So what do you mean? How many times do I have to ask you to define your terms?
The best lack all conviction
The worst are full of passionate intensity.
*
Oops, another double-post. Either that or the beginning of a finite recursion.
Hey Lib, you’re the one who claimed I was taking some sort of side because I asked you a simple question at the top of the thread.
Right now, a whole bunch of people who understand math far better than you are trying to get across the notion of infinity to you, and you’re ignoring them - not just me.
If my logic is so full of holes, pick it apart. I’ve corrected your proof; where’s the flaw? I made rational arguments; I contributed something substantive. If all you can offer in return is insults, well guess what? We’re talking mathematics here; invective scores no points.
If you don’t like my attitude, I don’t like yours, to be quite honest. For this thread, I think your only recourse is to put up with it: I’ve got the home-field advantage on this topic.
Mathematics isn’t a field where everyone’s opinion is equally valid. When you go around telling other posters things like “What I think doesn’t exist is a cardinality, A, for any field, F, where A + 1 = A. I find the concept about as useful as tits on a bull hog”, that’s nice, but a lot of rigorous work went into building up the concept of cardinality, including that of infinite cardinality, back in the nineteenth century. When you say, “I’m not unfamiliar with Cantor’s derivation of infinity from endpoints of line segments. I simply say that it’s a bunch of hooey,” it tells all of us is that you don’t give a flip what mathematicians have proved over the centuries; if it doesn’t meet your exacting standards, it’s irrelevant.
I’ll be waiting for your paper showing the flaws in the work of Georg Cantor. Have fun.
Spiritus, Cabbage, Dark Wing Duck, Gilligan, RM Mentock, Konrad, BeerUser, and others have all had it right, and done a good job explaining it to you. But hey, if you know better than them, me, and one of the most brilliant mathematicians of all time (Cantor, that is), then more power to you.
Just don’t blame it on my attitude: your willful ignorance on this subject will resolutely continue, by all appearances, well after I absent myself from this thread. Carry on as before.
Mother of God this is a classic thread! Thank’s Lib!
And thank you, Spiritus, for the most intellectually snide insult ever levied, which I plan on immediately incorporating into my repertoire of abuses:
Priceless!!!
I haven’t the background to add anything resembling substance to this thread, but I can’t refrain from asking a philosophical question: If conventional scientific knowledge postulates that the universe has a finite beginning and end, how is the concept of infinity anything but smoke and mirrors?
Hell is Other People.
Basically, Lib, I believe that your point is that there is no infinity, and you try to use two proofs that, using infinity, prove opposite things.
First, I think your concept of infinity is the main problem. IT IS NOT A NUMBER. Just so you know. Infinity is a concept used in limits and the like. In limits, when they say ‘as n approaches infinity’ they mean ‘as n gets really really big’. Or in ‘.9…’ it means ‘You know, there are a lot of nines there, I mean WOW that’s a lot of nines. You can go on and on and on counting nines and you won’t have enough. We’re talking major nineage. An infinite number of nines’. Or with lines, infinity is the right end marker of a thing with no right end.
You can’t really apply mathematical operations to infinity, because it’s not a number. You can, however, apply the concept of infinity to mathematical operations. That’s how limits work (approaching in infinitely small increments), and numbers like 2/7ths in decimal form, and proofs using .999…
Just remember, infinity is not a number. It is instead a concept vital to mathematics, theology, and philosophy.
Libertarian,
I was under the misapprehension that your example was misstated as a pedagogical device. I see now that it is misstated because you do not understand the meaning of the notation. Let us proceed with your proof, first, and worry later about the correct meaning of the notation of the mathematical expression of 1.999. . with a bar over the repeating portion of the decimal value.
Your notation .9[n] as presented is interpreted as 9 appearing n times after the decimal point, where nine appears only the number of times designated by a specific value of n. That makes your Premise 3 in proof one incorrect for each, and every finite interger.
Premise 3: 10X - X = 9.9[n] - .9[n]
Since in every case it should be: 10X - X = 9.9[n-1] - 0.9[n]
Or 8.99…1
The use of the convention for repeating decimals is not to approximate a “least finite real number” as you seem to think, but rather to address the limits of decimal notation in arithmetic in a manner that allows it to deal with rational numbers that don’t come out even.
&frac13 + &frac13 + &frac13 + = 1
.33… + .33… + .33… does not = 3
for any finite number of repetitions of 3. If you use a notation to indicate that the three repeats without end, then you must use it that way at all times, and never use it to imply some value n for the number of repetitions it has. There is no finitevalue of n that will provide the value of &frac13.
I am sorry I imputed curmudgeonry to what was simply a misinterpretation of the meaning of a mathematical convention.
We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about `and’. – **Sir Arthur Eddington, ** (1882-1944)
Darn,
Make that:
[sup]1[/sup]/[sub]3[/sub] + [sup]1[/sup]/[sub]3[/sub] + [sup]1[/sup]/[sub]3[/sub] + [sup]1[/sup]/[sub]3[/sub] = 1
Shouldn’t it also be:
.3… + .3… + .3… does not = 1
for any finite number of repetitions of 3.
(.33… + .33… + .33… does not = 3) works too, of course, but it seems somehow beside the point.
The best lack all conviction
The worst are full of passionate intensity.
*
Ok, so strictly speaking it should be:
0.33 (with an overline on the final three)
“Can you do addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?” “I don’t know,” said Alice. “I lost count.”
– Lewis Carroll, Through the Looking Glass
So many straw men, so little time.
Surgoshan:
Perhaps you will recall this line from the OP, rendered in all caps for your benefit: “AND REMEMBER, INFINITY IS NOT A NUMBER.”
Oh, limits. Well, why didn’t you just say so? [Eyes rolling…]
Just remember that I said it here before you did.
Well, theology and philsophy aside, the only reason you might reasonably call it “vital” to mathematics is its service as a cop-out for calculating the areas under curves and such.
RTFireFly:
That’s exactly the kind of intellectual arrogance I’m talking about. The hubris. You’ve made a parsimonious assumption based on nothing.
I’ve got four books beside me, and 15,000 web pages accesible to me, that all tap-dance around the notion of infinity. Thanks anyway.
The flaw? 
Aside from its utter incomprehensability, you mean? Besides, in typical fashion, you did not respect the OP by looking for flaws in the proofs given, but rather, evidently unable to help yourself, reeled off a “proof” (that did not even postulate that A = A) in direct opposition to the OP’s request to hold off on that.
RM Mentock:
You are exactly right. Thanks.
But I am not the author of that proof. (I just cleaned it up a little.) It has been commonly offered many times before. I am, however, the author of the second proof.
Correct again.
What do you mean by “an infinite number”? If A + 1 = A, then A is meaningless. So is 1.
In finite arithmetic, the largest number, of course, is the muliplicative inverse of the smallest one. Thus, the largest number is 1 / 0.0[n-1]1. If we call that number A, then A + 1 is undefined.
Which is as it should be. A + 1 does not equal A. It is simply as meaningless as A / 0.
Dark Wing Duck:
The number n is equal to the number of decimal places in the expression 0.0[n-1]1.
Sake:
You just made the effort worthwhile. Thanks. Even if people disagree, they don’t have to cop the attitude that I’m some moron.
I don’t think he intends it, but he does that to me all the time. If I define something for him, offer illustrations, and compose fifteen posts talking about it, his typical response will be, “But when are you going to define it.” The Atheist Religion thread is a litany of these.
But I like Spiritus. Very much, in fact. At least he doesn’t mean to be insulting.
That’s all it is. My message to others is, “Infinity does not exist. Get over it.”
The reason you cannot calculate the area under a curve on a Cartesian plane is that the coordinates are laid out in digitally discreet columns and rows, just like a spread sheet. Naturally, an analog shape, like a parabola, does not lend itself to measurement on that plane.
Limits (like the limit of x as x approaches infinity) merely testify that the universe is not a Cartesian plane. And they are very, well, limiting.
Tris:
Your contributions are always welcome, because they are always respectful and sincere. But I do understand the meaning behind the convention. I am only saying that the convention is myopic.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Cabbage:
The point is as I have said already. I give it again for your benefit:
4(.7) = 2.8
4(.77) = 3.08
4(.777) = 3.108
4(.7777) = 3.1108
4(.77777) = 3.11108
By saying that 4(.7…) equals 3.1…, you are mystically throwing away not just one, but two — two! — significant places, or else you are rounding up.
4(.7[n]) = 3.1[n-2]08.
How do you account for the two significant places in 3.1… that somehow were pushed out of the universal set?
“It is lucky for rulers that men do not think.” — Adolf Hitler
Libertarian, again I ask, where in the world did you get the idea that there is some sort of struggle between “infinists” and “anti-infinists”? Are there mathematical textbooks out there debating the the subject, because I had never heard those terms before you brought up this supposed Great Debate.
BTW, we do not have to automatically respect the OP if said OP doesn’t make any sense. Or haven’t you noticed that you haven’t exactly got a flood of support for your premise, let alone your questionable matematics. If you want to disprove such a universally accepted concept as Infinity, you’ll have to do a lot better than this.
Now I know how Galileo felt. “You mean you seek to overturn centuries of tradition held by our greatest thinkers!?”
You don’t seem to read very carefully. The OP didn’t say that there is a debate between infinitists and anti-infinitists, but merely that one might develop here.
It also noted that there would likely be many more infinitists than anti-infinitists because most people would simply rather not rethink something. They are like Bible thumpers: Cantor said it. I believe it. That settles it.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Guys, I think Libertarian is just putting us on.
But, just in case:
Infinity does not exist, it is just a useful concept.
Cartesian planes are not discrete. A simple sine curve will have values on the y-axis which are irrational. And, also, you can easily calculate most areas under a curve by integration.
There’s always another beer.
Cantor proved it. Better minds than ours have verified the rigorousness of his logic. We can do the same, or try to find the flaws in his reasoning, if we want to.
This is how mathematics is done. We prove stuff. Before our proofs are published in refereed journals, other people go over our proofs to make sure that they are valid, that there are absolutely no logical missteps. And once a proof is published, everyone else in the world is free to go over the proof as well, to ascertain its correctness or find error.
I haven’t done that with much of Cantor’s work, specifically, other than the basics that every Ph.D. mathematician gets before s/he passes his/her quals, before we’re allowed to specialize. But I’ve done that with the proofs in my area that I rely on, in order to build from there, and so does every other mathematician.
I’m a graph theorist, not a set theorist. But if I were a set theorist, I’d be able to tell you that I’d taken exactly none of Cantor on faith. Since there are plenty of set theorists out there, I am content to rely on their having not taken Cantor on faith.
Because none of them has managed to find any flaws in his work, that settles it, until someone does.
Argument dead. As I understand it, what Lib is doing falls under the definition of trolling: supposedly starting a debate, but actually taking one side of it and refusing to give bona fide consideration to opposing points of view.
In this instance, it was never a debate, since in mathematics, as I’ve said, we don’t debate: we prove or disprove. But the principle holds. Lib, quit trolling.