Libertarian
All right! I believe that your system may be consistent. I’m not sure how useful it is, though.
If 3.1[n]/4 is undefined, then it would seem that 3.1[n-2]10/4, and 3.1[n-2]09/4 are also undefined in your system, but 3.1[n-2]08/4 would equal .7[n]. Is this correct?
Whenever you tried to do division, you’d have to go all the way out to the end to see if it stopped properly, or else it would be undefined. Seems like a lot of work.
Patience. Notice, they appear above.
.
Libertarian:
n is a number, by your definition. It is either your hypothetical largest number, or it is not. In either case, n-1 is not the largest number. So I can add 1 to n-1 to get n. There is one more 0 to the right of the decimal point in 0.0[n]1 than in your proposed smallest number, 0.0[n-1]1. Therefore 0.0[n]1 is less than 0.0[n-1]1. Therefore 0.0[n-1]1 is not the smallest number.
(slaps his head)
RM Mentock:
Okay, you have me hooked. Reel me in.
Let’s see. 3.1[n-2]11 is equal to 3.1[n]
3.1[n-2]10 is one-tenth of one magnitude smaller than 3.1[n]
3.1[n-2]09 is one-tenth of one magnitude smaller than 3.1[n-2]10
3.1[n-2]08 is one-tenth of one magnitude smaller than 3.1[n-2]09
Therefore, 3.1[n] is greater than 3.1[n-2]08 by three-tenths of one magnitude.
Therefore, .7[n] + .0[n-2]03 = 1
Is that correct?
“It is lucky for rulers that men do not think.” — Adolf Hitler
Because of Zeno, infinity got a bad rap. Until Cantor showed that infinity can be talked about as a concept, “potential” infinities were allowed but not “actual” infinities. This is the reason limits were introduced into calculus. There is a foundation for calculus called non-standard analysis that allows infinitesimals (and their inverses infinities). Non-standard analysis can be shown to be consistent(sp?) (well,… as consistent as standard ananysis)
If the last natural number is 10[n-1], what is 10[n-1] + 1? It must a number because of the closure of addition. It is trivial to show it is greater than your largest number.
What happened to the two significant digits in 3.1…08? Nothing happened to any significant digits. The two significant digits are 3 and 1. the zero and the 8 are the two least significant digits.
What about:
1/3 = .33333…
1/3 + 1/3 + 1/3 = .99999…
If 1 does not equal .9999… then the whole is greater than the sum of its parts???
Note that if you use base nine instead of ten, 1/3 + 1/3 + 1/3 = .3 + .3 + .3 = 1.0
The laws of arithmetic depend on the base used for notation!!! (NO, I don’t think so!)
I also take issue with infinity is not a number. It depends upon what system of numbers you are talking about. A number can be whatever you define it as. One half is not a number if you only consider integers. Sqrt(-1) is not a number if you are talking about reals. Infinity is not an integer. OK. Infinity is not a real number. OK. However if you talk about transfinite numbers, then you have plenty of infinities.
I think rather than demonstrate that infinity is not a number, you have shown that you do not understand the concept of infinity. (or rather (just to be kind) that you take the pre-Cantor stance.)
Virtually yours,
DrMatrix
Spiritus:
If they are not missing, why do they not appear in your solution?
Firefly:
I’m glad you’re sincere. Please be mindful of people’s feelings in the future. You have no right to assume people know less than you do simply because they ask questions or raise debates.
No. It means a 1, followed by n-1 zeros.
As I said, it is the multiplicative inverse of the smallest number, which itself is 1 - .9[n]. The multiplicative inverse of the smallest number is by definition the largest number.
I don’t understand that question.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Hmmm, are you sure this isn’t a religious thread? I think you’re trying to sneak it in towards the end Lib. 
An interesting system you’re setting up, but I don’t think I see how it can be more useful than the one we normally use. Also, as someone stated, I think you will have to explain how/why addition is no longer closed.
PeeQueue
PeeQueue:
Hey, aren’t you the guy I’m taking all this heat for?! See whether I start another nonreligious nonpolitical thread!
“It is lucky for rulers that men do not think.” — Adolf Hitler
Hehe, yeah I do feel a little guilty for that. I don’t know why everyone got so riled up about this. Not that I agree with you or anything…
PeeQueue
Dr. Matrix, that was succinct, lucid and comprehensive. Thanks for the best post on this thread so far.
I was thinking of where you said:
So if 10[n-1] is the last natural number, that raises the question, for what (lesser) natural number, n, is this so? (How many times does the zero repeat?)
Alsø, what set of axioms are we operating under? Those preceding Proof 1? Proof 2?
Libertarian
Naw, you didn’t bite. I’m going to have to yank ya. You were supposed to admit that 3.1[n-2]10/4 is undefined, just like you said 3.1[n]/4 is undefined. Why are they undefined? Because they don’t equal some decimal that terminates at [n] places.
Thus, every fraction that doesn’t terminate, every fraction whose denominator has factors other than 2 and 5, is undefined. One third is undefined, in your system. One sixth is undefined, one seventh, one ninth.
How come? Because you threw them away when you built your system.
So, we can now look at the “08” problem.
4(.777) = 3.108
4(.7777) = 3.1108
…
4(.7[n]) = 3.1[n-2]08
You ask, where does the “08” go when we say that 4(.7…) = 3.1…
In our system .7… equals 7/9, so 4(7/9) equals 28/9, which in our system equals 3.1…
The “08” never shows up. We don’t throw them away, they were never there in the first place. Our system makes a concerted effort to keep as much as possible. We don’t like to throw away (make undefined) nice simple fractions like 1/3 or 1/7. We keep them.
Notice the following:
4(7[n]) should equal 31[n-2]08, but it does not because 31[n-2]08 is undefined. In our system, we keep the product, instead of making it undefined.
.
You again beg the question of the value of n.
“n is the elargest possible integer.
n is defined as the inverse of the smallest possible number.
The smallest possible number is 1 - .9[n].”
Can you see the circle?
The best lack all conviction
The worst are full of passionate intensity.
*
Lib - are you just trying to prove infinity isn’t necessary in ‘normal’ mathematics? Or are you trying to come up with a whole new axiomatic system?
Libertarian posted 01-09-2000 05:27 PM
Perhaps that’s your problem. You seem to think that when people speak of the cardinality of a set, they’re talking about a number. Some cardinalities can be represented by number, but not all. The cardinality of the set of real numbers, for instance, can not be represented by a number. There is no cardinality A such that A+1=A.
Libertarian posted 01-10-2000 06:12 AM
And how many decimal places are there? n! Aren’tcircular definitions fun?
Libertarian posted 01-10-2000 08:12 AM
Perhaps you would like to actually tell us what you mean by .7…?
Libertarian posted 01-10-2000 08:18 AM
RM Mentock posted 01-10-2000 11:36 AM
Libertarian posted 01-10-2000 11:54 AM
RM Mentock:
quote:
I went back through all the posts in this thread, and I could not find those reasons. I even searched for “undefined”. Could you give me the date of the post? Thanks.
01-10-2000 6:12 AM Central
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.
[/quote]
RM Mentock askedd about where you had said that n was undefined, so you give a quote about how 1/0.0[n-1]1 + 1 is undefined? That doesn’t make sense, except maybe in your world. What makes you think you can make up a completely new type of notation, never define it, and expect people to understand it?
Libertarian posted 01-10-2000 01:35 PM
Asking questions or raising debates I have no problem with. Not defining terms, claiming that one has defined terms, and then complaining when people insist that one define one’s terms and pretending that on is being harassed I do have a problem with.
BTW, your sig gets more ironic the more times I read it!
Lib, I’m getting confused. You’re trying to prove there is no infinity, and say there is a largest number?
Libertarian:
Do you see the contradiction? By Axiom 5, A + 1 = A - (-1) IS defined. By the latter quote, A + 1 IS NOT defined.
Another point I wanted to make: Suppose S is the smallest positve real, L the largest.
What is the order of the following numbers: S, L, S/2, L/(1/2)? How about L/S or S/L? I’m asking because you haven’t really developed the notion of order and how it relates to the arithmetic you’re using (I’m not being sarcastic, I’m trying to go somewhere with this).
Libertarian, I owe you an apology. Here I’ve been tossing around expressions involving n without realizing, as should have been clear from the context, that n is the greatest integer. An expression such as 0.0[n]1 is meaningless, as that would imply that there were n+1 digits to the right of the decimal point.
I particularly appreciate your identification of the largest integer as 10[n-1], since that allows us to calculate the exact value of n. In fact,
10[n-1] = n
implies (taking the base 10 log of both sides)
n-1 = log(n)
from which any fool can see that the greatest integer is 1. The smallest positive number, 0.0[n-1],1 is 0.1, and the largest number is 1/0.0[n-1]1 = 10. This may be related to the fact that people have ten fingers, and “man is the measure of all things.” Let no one suppose, however, that 10 is an integer, since integers greater than 1 do not exist. This new mathematical system should come as a relief to everyone - no more worrying about those pesky numbers bigger than 10, or very small fractions. I’m certainly going to sleep better at night.
Libertarian wrote:
Considering that the usual Fundamentalist Christian view is that the universe is less than 10000 years old, it’s amazing to me that a Fundamentalist Christian could buy into infinity at all.