Actually, Lib, I’m not versed in number theory at all, and the implications of the equations totally escape me at the moment.
I am, however, looking forward to the exchange regarding [spooky voice] what it all means. [/spooky voice]
Or just watching another bull session between you and Spiritus. 
Konrad:
That’s what I say!
Infinitists have gotten away for way too long with saying, on the one hand, “infinity’s not a number,” and on the other hand, “there is an infinite number of nines”.
The problem with Premise 2, of course is that it should read: 10X = 9.9[n-1], else we have ghosts and lawn sprites in our digits.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Quixotic:
Well, the implication of Proof 2 is that the set of real numbers is finite, and that there is both a smallest number in magnitude ABS((0.0[n-1]1)) and a largest ABS((1 / (0.0[n-1]1)))! It also implies that 0.0[n-1]1 is next to 0 on the real number line — with no other number between.
It is the anti-infinity proof.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Where’s tracer? I just know that he is an anti-infinitist.
“It is lucky for rulers that men do not think.” — Adolf Hitler
“Definition: .9… = .9[n].”
“And remember, infinity is not a number.”
"…I used [n] to indicate the number of times a digit repeats. "
What is the value of n?
Lib:
X divided by 0 is mathematically undefined.
0[n-1]1 is logically undefined in your proof. The notation you used to define 9[n], as I described before, is universally accepted to mean an unending repetition of the same digit(s). If you untended it to mean something else then you should have specified that. “One less than an unending sequence” is a meaningless phrase. An unending sequence has no ordinal value to take one less than.
BTW – the same problem holds for your restatement of premise 2 for proof one. You are correct that the phrase “an infinite number of [anything]” lacks rigorous meaning. It is simply a convenient shorthand which is useful only in non-formal contexts. You will note, I trust, that I have not used it. Unending repetition of a sequence, however, contains no such paradox. That is also how the notation 9… is interpreted. Nowhere in your proof did you say that 9… meant an “infinite number” of 9’s. Nowhere in my analysis did I assume the same.
The best lack all conviction
The worst are full of passionate intensity.
*
I didn’t side with anything, Lib. I was trying to get you to define your [n] notation, that’s all. Which you eventually did:
Now let’s take your ‘proofs’ to see what (if anything) you’ve shown:
Not very interesting result.
For proof 2, all we need are the concluding steps:
Which are absolutely true, given your clarified definition of the [n] notation, and don’t contradict the corrected Proof 1 in the least.
The only contradiction here is if you take .999… (or a similar term) to mean 9’s for an infinite number of places in one proof, but not in the other. That’s what you did. But that’s not really a contradiction; it’s the fallacy of equivocation - using one term to mean two different things in the same argument.
Glad I didn’t throw away my logic texts when I quit being a math professor…
Dealing with faulty assumptions:
Not hardly. Infinity isn’t a number; it’s the concept of continuation without end. An infinite sequence or series has no last term, for example. So we’re not doing operations on infinity; we’re doing operations on the individual terms of the infinite series, multiplying each one by 10, for instance.
For what .999… is, is an infinite series. That’s its definition: it means 9/10 + 9/10^2 + 9/10^3 + … ad infinitum, quite literally. Multiply it by 10, and you get 9 + 9/10 + 9/10^2 + 9/10^3 + … ad infinitum. (The 9/10^3 in this series, is 10 x the unwritten 9/10^4 term from the previous step.)
I’m not sure what one has to do with the other. The universal set - the set of real numbers - isn’t just infinitely large, but it’s too big to be put into 1-1 correspondence with smaller infinite sets such as the integers or the rationals.
But that doesn’t mean that some of those infinitely many numbers (meaning, again, that the supply of such numbers continues without end, not that ‘infinity’ is a number) are between .999… and 1.
And, btw, if A and B are unequal real numbers, it isn’t an assumption that there’s a real number between them; it’s one of those things that’s easy to prove from the basic axioms, though. Maybe later.
OK, it’s later already.
Axiom 1: if A<B, then A+C < B+C.
Axiom 2: if A<B and C>0, then A/C < B/C.
Axiom 3: if n is a positive integer, then A = A/n + A/n + … + A/n, with n identical terms in the sum.
(Coulda probably proven that from something more basic, but I figured we’d agree on it.)
Theorem: If A<B, then there is a number between A and B.
Proof: Assume A<B. Then A/2 < B/2.
Then A = A/2 + A/2 < A/2 + B/2 < B/2 + B/2 = B.
A/2 + B/2, being greater than A and less than B, is between A and B. QED.
I’ve corrected a lot of math homework in my life, but this is the first time I’ve done it over the Internet!
I screwed up; I forgot that the inequality signs, in opposite directions, can cause stuff between them to disappear. Big oops!
So I’m going to use ‘{’ and ‘}’ in lieu of inequality signs; it’s the easiest fix.
Axiom 1: if A{B, then A+C { B+C.
Axiom 2: if A{B and C}0, then A/C { B/C.
Axiom 3: if n is a positive integer, then A = A/n + A/n + … + A/n, with n identical terms in the sum.
(Coulda probably proven that from something more basic, but I figured we’d agree on it.)
Theorem: If, for real numbers A and B, A{B, then there’s a real number between A and B.
Proof: Assume A{B. Then A/2 { B/2.
Then A = A/2 + A/2 { A/2 + B/2 { B/2 + B/2 = B.
A/2 + B/2, being greater than A and less than B, is between A and B. QED.
There; that should come out right, even if it looks funny.
Oh yeah. I could prove that there’s nothing in between .999… and 1, but that’s an epsilon proof. (As in, “let epsilon > 0…”, that phrase that has intimidated generations of calculus students.) You don’t want to mess with epsilons.
Lib,
An infinite number of zeros, followed by a one? Please. You can’t be serious, can you? Good arithmetic makes sense when you say it out loud.
Mama T, my ancient, not-so-sainted basic algebra teacher taught me that infinity means “I don’t know, but it seems like it ought to be real big.”
[grumble] Infinite series followed by a blind man. [/grumble]
The different branches of Arithmetic – **Ambition, Distraction, Uglification, and Derision. – **Lewis Carroll, ** Alice in Wonderland.
Libertarian:
Yeah, and anti-Ptlomics have gotten away all this time with referring to “sunrise” even though they insist that the sun doesn’t move.
The sentence “a decimal point followed by an infinite number of nines is equal to one” is, strictly speaking, false. So is the sentence “a decimal point followed by an infinite number of nines is less than one.”
“A decimal point followed by an infinite number of nines” is, stricly speaking, no more a real number than three watermelons balanced on top of a chair is a real number. “A decimal point followed by an infinite number of nines” is shorthand for “The limit as n goes to infinity of a decimal point followed by n number of nines”. If take a shorthand notation beyond what the situations it’s supposed to be used, you’ve going to end up with contradictions. If you look at the rigorous definitions, however, you will see no contradictions. Infinity isn’t a real number, but it’s used as a real number in the same way that dy/dx is used as a fraction, even though it’s not.
That should be “If you take a shorthand beyond the situations in which it is supposed to be use, you’re going to end up with contradictions.”
Well, I’m not sure what you’re trying to say. If I hold in my mind the idea of an unending string of numbers, and decide to call that idea infinity, how can someone tell me there is no such idea?
I think I stand on the side of .0[n-1]1 is meaningless. If I’ve decided that an unending string of things, the idea of which I will call infinity, shall be represented by the symbol [n], then I must realize that taking one thing out of that string will still leave me with an unending string. So, how do you stick a 1 at the end of the unending string of stuff? It doesn’t make sense.
And besides, why would you want to do away with the concept of infinity after all the great new things it has enabled mathematics to do?
Interesting stuff though.
PeeQueue
Libertarian, having read your equation, I have no idea what you are talking about. I just tossed the whole equation out the window when you tried to divide by zero, and came up with a definition of Infinity, which cannot be proven or disproven with the faulty equations you provided. On the other hand, if you think that you can disprove the concept of Infinity with a mathematical formula, why don’t you give a call to M.I.T. or Cal Tech. They love calls of that sort.
Konrad - sorry for not noticing that you beat me to the punch:
and
Good going!
Damn it! I’m late. Spritus seems to have got it.
The flaw is in:
Premise 7: 10 - .9[n] = .0[n-1]1 (by Axiom 5)
Premise 8: .0[n-1]1 > 0 (by Axiom 4 and Premise 1)
In proof #2
The expression .0[n-1]1 should be equal to zero as n goes to infinity. The expression .0[n-1]1 can be better written as 1/10^(n+1) (shame on you, lib) which clearly goes to zero as n goes to infinity.
But upon further consideration, you make no mention of infinity. If these are just discrete numbers, then the flaw would be in proof 1, premise 2:
10X does not equal 9.9[n] but rather 9.9[n-1]
Do I get the prize or am I too late?
There’s always another beer.
Looks like RTfirefly beat me too.
Do I get a consolation prize, or something?
There’s always another beer.
Consolation prize: stop by for your choice of Beck’s, Corona, or Negra Modelo.