That would PROBABLY be because the arrow DIDN'T perform to that model? In which case, the example is moot?xouper said:So - since the arrow does in fact touch you, there must be something wrong with analyzing the number the way Zeno would have you analyze it.
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xouper said:The notation 0.999... is simply shorthand for the infinite series (0.9 + 0.09 + 0.009 + ... ). They are different notations for the same exact thing. Does this answer your question?
Thank you for confirming this, Xouper. So...
We have an infinite series that sums to a value that, by induction, approaches but does not equal 1. This same infinite series has a limit value of exactly 1. And this same infinite series is a notation for 0.9 (rec), as confirmed above. ?????
I can only take it that the properties of this infinite series changes somehow at the point that the number of terms becomes infinite. wEiRd !!
Well, I wish I knew a nice way to say this so please pardon my directness here, but in mathematics, sloppy use of jargon implies a sloppy understanding of the concepts. In other words, when you use the jargon incorrectly, it is difficult to tell whether your misunderstanding is with the jargon or with the underlying concepts.
Please be direct - I expect no less. And you are quite right about my presentation, and I make no claim to being a totally competent mathematician. My apologies if I'm confusing one and all. However the CONCEPT I'm driving at is fairly clear in my head, so I'm trying to get it out and have it criticized properly. Peer review and all that!
This is informative. Thanks, Cabbage.
I see where you are coming from now! I have much to think on here...Infinitesimal and infinite numbers
A nonstandard real number e is called infinitesimal if it is smaller than every positive real number and bigger than every negative real number. Zero is an infinitesimal, but non-zero infinitesimals also exist: take for instance the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...) (this works because U contains all index sets whose complement is finite).
A non-standard real number x is called finite if there exists a natural number n such that – n < x < +n; otherwise, x is called infinite. Infinite numbers exist; take for instance the class of the sequence (1, 2, 3, 4, 5, ...). A non-zero number x is infinite if and only if 1/x is infinitesimal.
Now it turns out that every finite nonstandard real number is "very close" to a unique real number, in the following sense: if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x. This operation has nice properties:
* st(x + y) = st(x) + stif both x and y are finite
* st(xy) = st(x) st
* st(1/x) = 1 / st(x) if x is finite and not infinitesimal.
* the map st is continuous with respect to the order topology on the finite hyperreals.
* st(x) = x if and only if x is real
I'm not claiming it as a proof or disproof of anything really, just following my own logic through. I'm in an unenviable position of studying some but not taking a degree in pure math - I can follow the arguments in most higher math, but I'm certainly not practiced enough to construct proper arguments using the proper terms very proficiently.Cabbage said:
To the question: Yes, I think you have stated my position fairly well. No, I can't see the flaw just now. No, I don't mind at all being shown it - I'm not woo-woo.
I fully agree. But I'm not advocating stopping the series at any point in the finite set but continuing it to infinity. And at that point its properties change...Cabbage said:
An interesting concept: for any finite number of terms up to but not including infinity, the sum of the series is not-quite-1. For an infinite sum, it is exactly-1. So how big is the finite set before it becomes infinite? Infinitely big? Just less than infinitely big?
The arrow does the same exact thing that 0.999... does. Just as the arrow travels the entire distance and hits you, 0.999... "travels" the entire distance and equals 1. The only difference between the arrow and 0.999... is that the arrow takes non-zero amount of time to travel, whereas 0.999... is there in zero time.
No. The infinite series is exactly equal to one. This equality is established by the geometirc series theorem.
No. The infinite series is equal to one. The infinite series does not have a limit. Perhaps what you are thinking is that the infinite sequence of partial sums has a limit, and you would be correct. But the series itself is exactly equal to one.
Yes.
Cabbage addressed this question rather well with the example of the natural numbers. At what point does the set of natural numbers become infinite?
Let's consider the arrow example again. When I shoot an arrow at your head, somewhere along the way, it changes from "not quite there" to exactly there. How did that happen? It happens when you stop considering it as a set of smaller and smaller steps along the way, and take the distance as a whole.
Likewise, when you stop considering 0.999... as a set of smaller and smaller steps along the way, and take the entire distance as a whole, then you will have changed from "not quite there" to exactly equal to one.
A set is either finite or infinite. You can't make a finite set infinite by making it "infinitaly big". Even if you define your finite set to have 'n' elements, it is still finite. It doesn't matter how big 'n' is. And even though there are infinite possibilities for 'n', it is still a finite set with 'n' elements.Zep said:
First, there is no before or after. A set is either finite or infinite. A finite set cannot "become" infinite; the inifinite set is a different set altogether.
This has been mentioned before, but it is worth repeating. The natural numbers is a set. There is an infinite number of them, and we can play with infinity in this set, but you can't "get" there in the way you describe. In other words, you can't count to infinity. If you are there, you are there. You can't "get" there.
Cabbage
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Zep, it might help in understanding if we extend the discussion to the ordinal numbers.
No doubt you're familiar with the first ordinals--they are the natural numbers 0, 1, 2, 3,....
There's a standard set theoretic construction of the ordinals (due to von Neumann) which may help in visualizing them (it probably won't help visualizing the finite ones, but it may help for the infinite ones):
1. 0 = {} (the empty set).
2. If n is an ordinal, then define n+1 = n union {n}.
In other words, 2 can be rephrased as saying, "A given ordinal is equal to the set of smaller ordinals".
The first few ordinals:
0 = {}
1 = {0} = {{}}
2 = {0,1} = {{},{{}}}
3 = {0,1,2} = {{},{{}},{{},{{}}}}
and so on.
So far, we've defined what the first ordinal is, and we've got the successor ordinal (n+1).
If that was all we had, we could never get an infinite ordinal--you can't reach infinity by starting with zero and adding one over and over again.
We need a new axiom:
The axiom of infinity: There is a set containing 0, and whenever n is in the set, n+1 is also in the set.
w (omega) is defined to be the smallest such set (the set of natural numbers):
w = {0,1,2,3,4,...}
Without the axiom of infinity, the "set" of natural numbers wouldn't actually be a set at all--there's no way to construct it otherwise.
Once you get that first infinite set, however, all hell breaks loose, pretty much. Going back to 2. above, we get:
w+1 = w union {w} = {0,1,2,3,...,w}
w+2 = w+1 union {w+1} ={0,1,2,3,...w, w+1}
w+w = {0,1,2,3,...,w,w+1,w+2,w+3,...}
w+w+1 = {0,1,2,3,...,w,w+1,w+2,w+3,...,w+w}
And so on, going on to ordinals with cardinality larger than w (=the set of finite ordinals (natural numbers)), such as w<sub>1</sub> (=the set of countable ordinals), w<sub>2</sub>,...,w<sub>w</sub>, and so on.
All ordinals are well-ordered sets, meaning any (nonempty) subset of an ordinal has a smallest element (but not necessarily a largest element--w has no largest element for example (there is no largest natural numbers)). The collection of ordinals itself is not a set (it's "too big"--it can't be assigned a cardinality), but the proper class of all ordinals itself is "well-ordered", as well.
You might notice here that there are two types of ordinals. 3, for example, is equal to 2+1. w, on the other hand, can't be written as x+1 for any ordinal x--there is no ordinal immediately before w. This is basically another way of saying, "There is no 'point' at which the natural numbers become an infinite set--if you cut them short at any point, you only have finitely many".
Ordinals of the form x+1 for some x are called successor ordinals, other (non-zero) ordinals are called limit ordinals. We can think of a limit ordinal as being the union of all the ordinals preceding it (w, for example, is the union of all natural numbers).
Now, back to my previous fallacious induction argument. There are two types of induction:
1. Standard induction: P(0) and (P
implies P(n+1)) together imply P(m) for all natural numbers m.
2. Transfinite induction: P(0) and (P for all n < m implies P(m)) together imply P(x) for all ordinal numbers x.
Type 1 can only handle "jumps" of n+1--it can't make the jump to w, since it isn't of the form n+1. My original proof was of type 1--that proof demonstrated that .999...9 is not equal to 1 for any finite number of 9's, but can't make that "jump" to say anything about the case with an infinite number of 9's.
If a similar proof of type 2 could be constructed, however, that wouldn't be a problem. Type 2 just relies on the well-ordering property of the ordinals--it doesn't require the jumps to be made at successor ordinals only, it can handle limit ordinals as well.
Type 1 always breaks down at omega, since omega is the first limit ordinal.
I don't know if any of that really helps at all; this is a huge topic that can be difficult to digest, and difficult to condense into a nutshell. There's so much we know (and much we don't know) about ordinals, but I figure I ought to stop here.
No doubt you're familiar with the first ordinals--they are the natural numbers 0, 1, 2, 3,....
There's a standard set theoretic construction of the ordinals (due to von Neumann) which may help in visualizing them (it probably won't help visualizing the finite ones, but it may help for the infinite ones):
1. 0 = {} (the empty set).
2. If n is an ordinal, then define n+1 = n union {n}.
In other words, 2 can be rephrased as saying, "A given ordinal is equal to the set of smaller ordinals".
The first few ordinals:
0 = {}
1 = {0} = {{}}
2 = {0,1} = {{},{{}}}
3 = {0,1,2} = {{},{{}},{{},{{}}}}
and so on.
So far, we've defined what the first ordinal is, and we've got the successor ordinal (n+1).
If that was all we had, we could never get an infinite ordinal--you can't reach infinity by starting with zero and adding one over and over again.
We need a new axiom:
The axiom of infinity: There is a set containing 0, and whenever n is in the set, n+1 is also in the set.
w (omega) is defined to be the smallest such set (the set of natural numbers):
w = {0,1,2,3,4,...}
Without the axiom of infinity, the "set" of natural numbers wouldn't actually be a set at all--there's no way to construct it otherwise.
Once you get that first infinite set, however, all hell breaks loose, pretty much. Going back to 2. above, we get:
w+1 = w union {w} = {0,1,2,3,...,w}
w+2 = w+1 union {w+1} ={0,1,2,3,...w, w+1}
w+w = {0,1,2,3,...,w,w+1,w+2,w+3,...}
w+w+1 = {0,1,2,3,...,w,w+1,w+2,w+3,...,w+w}
And so on, going on to ordinals with cardinality larger than w (=the set of finite ordinals (natural numbers)), such as w<sub>1</sub> (=the set of countable ordinals), w<sub>2</sub>,...,w<sub>w</sub>, and so on.
All ordinals are well-ordered sets, meaning any (nonempty) subset of an ordinal has a smallest element (but not necessarily a largest element--w has no largest element for example (there is no largest natural numbers)). The collection of ordinals itself is not a set (it's "too big"--it can't be assigned a cardinality), but the proper class of all ordinals itself is "well-ordered", as well.
You might notice here that there are two types of ordinals. 3, for example, is equal to 2+1. w, on the other hand, can't be written as x+1 for any ordinal x--there is no ordinal immediately before w. This is basically another way of saying, "There is no 'point' at which the natural numbers become an infinite set--if you cut them short at any point, you only have finitely many".
Ordinals of the form x+1 for some x are called successor ordinals, other (non-zero) ordinals are called limit ordinals. We can think of a limit ordinal as being the union of all the ordinals preceding it (w, for example, is the union of all natural numbers).
Now, back to my previous fallacious induction argument. There are two types of induction:
1. Standard induction: P(0) and (P
implies P(n+1)) together imply P(m) for all natural numbers m.2. Transfinite induction: P(0) and (P
for all n < m implies P(m)) together imply P(x) for all ordinal numbers x.Type 1 can only handle "jumps" of n+1--it can't make the jump to w, since it isn't of the form n+1. My original proof was of type 1--that proof demonstrated that .999...9 is not equal to 1 for any finite number of 9's, but can't make that "jump" to say anything about the case with an infinite number of 9's.
If a similar proof of type 2 could be constructed, however, that wouldn't be a problem. Type 2 just relies on the well-ordering property of the ordinals--it doesn't require the jumps to be made at successor ordinals only, it can handle limit ordinals as well.
Type 1 always breaks down at omega, since omega is the first limit ordinal.
I don't know if any of that really helps at all; this is a huge topic that can be difficult to digest, and difficult to condense into a nutshell. There's so much we know (and much we don't know) about ordinals, but I figure I ought to stop here.
gentlehorse
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Zep,
How about this:
The solution is that it's possible for the arrow to occupy infinitely many positions during a finite length of time. The distance traveled by the arrow and the time required for the arrow to reach its target form an infinite geometric series that has a finite sum.
Any help? If not, sorry for the interruption.
How about this:
The solution is that it's possible for the arrow to occupy infinitely many positions during a finite length of time. The distance traveled by the arrow and the time required for the arrow to reach its target form an infinite geometric series that has a finite sum.
Any help? If not, sorry for the interruption.
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Zep said:
Zep,
There's nothing "vague" about an infinitely repeating series. It's very well-defined.
I mean no offense, but your remarks suggest that you don't understand the concept of infinite. It means unbounded, or existing beyond or being greater than any arbitrarily large number.
Infinity does f*ck with your head, no doubt. It appears throughout physics and yields very weird results, especially in general relativity. Nevetheless, it's a real concept and there's nothing vague about it.
Of course, infinite and infinity appear throughout advanced mathematics too.
1/3 does in fact equal .333...
There is nothing vague about it. It's not an approximation, as you seem to imply.
3/3 equals .999.... equals 1. It equals 1 just as much as 1 x 1 = 1, or 3 - 2 = 1.
There is no approximation or "for all intents and purposes" involved. "Equals" here means "exactly equals."
Mathematicians would also say .999... and 1 are equivalent. One can substitute one or the other and mean exactly the same thing.
Just accept it. You will fail in any attempt to disprove or find fault with the proof offered throughout this thread. There simply isn't any "trick" involved in it and there's no approximation involved. It is what it is, and it is one hundred percent correct.
AS
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xouper said:I blame Zeno.
Xouper,
It took me a while, but I just now got this.
I agree, for those with some higher math. Those without even some higher math probably have never heard of his paradox.
AS
OK, first, to all who have been trying to help me understand: Xouper, Cabbage, patoco12, AmateurScientist and others, thanks for your input. It is indeed most informative.
But I think I need time to get my head around the presented information now. The old noggin isn't as sharp as it used to be on this, and I can feel the rust clogging the gears...
Xouper, I will try harder next time to use the correct terms! {note to self: STUDY!}
Cabbage: First pass of your information is making SOME sense to me in that I am following the arguments generally, but I'm going to need some time to assimilate it properly in detail. Your previous link to Wikipedia was helpful too. {note to self: MORE RESEARCH!}
AS: Your last post, on its own, sounds like being told by a woo-woo that "your ideas are wrong, you must believe me instead, you must not question me, you MUST accept my theory". But I know you didn't mean it that way, so I take it as a prompt to try to assimilate this higher math that I have been confronted with, and not just ignore it.
*sigh*
And to think I thought this was all behind me when I graduated...
But I think I need time to get my head around the presented information now. The old noggin isn't as sharp as it used to be on this, and I can feel the rust clogging the gears...
Xouper, I will try harder next time to use the correct terms! {note to self: STUDY!}
Cabbage: First pass of your information is making SOME sense to me in that I am following the arguments generally, but I'm going to need some time to assimilate it properly in detail. Your previous link to Wikipedia was helpful too. {note to self: MORE RESEARCH!}
AS: Your last post, on its own, sounds like being told by a woo-woo that "your ideas are wrong, you must believe me instead, you must not question me, you MUST accept my theory". But I know you didn't mean it that way, so I take it as a prompt to try to assimilate this higher math that I have been confronted with, and not just ignore it.
*sigh*
And to think I thought this was all behind me when I graduated...
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Zep said:
Ha ha. Mathematics as "woo-woo." No, of course I didn't mean it that way.
Math is not nebulous. It uses well-defined axioms and theorems. Mathematical proofs are not subject to having subjective opinions about their correctness. They are either correct or not. They may be inelegant or verbose, or they may be beautiful and succinct, of course, but their correctness isn't subjective.
Your opinion about the proofs in this thread doesn't change their correctness. The matter of correctness simply isn't subject to debate. One can discuss the proofs, but no amount of criticism will change whether or not they are valid proofs.
Is it reasonable to be skeptical about whether 2 + 2 = 4, for instance? I would suggest that it is only if one does not understand the concepts represented by the conventional symbols used in the arithmetic cited. It is demonstrably true.
I think you've focused too much on the limit on a series concept without really understanding it. Perhaps it's the notion of infinitely repeating that trips you up. I haven't read the whole thread, mind you, but that's the impression I got from at least some of your posts in it. If I'm wrong about that, then I apologize.
Anyway, I hardly think you are a woo-woo, and I suspect you do not believe I am either. Skepticism about well-defined and thoroughly understood and studied math isn't reasonable, however. Well-developed math doesn't change much in updating its "theories" like science often does. There simply isn't any data gathering involved, so there won't be any earth-shattering news that will cause previously held theories to crumble. Math is subject only to further exploration and invention abstractly, not to more data gathering.
In math, we speak of axioms, theorems, and proofs. Axioms are things we take to be given. It is axiomatic that 0 + N = N, for instance. Theorems are propositions based upon axiomatic assumptions that are either demonstrably true or not. Here's an example; it's called the Mean-Value Theorem.
Let f(x) be differentiable on the open interval (a, b) and continuous on the closed interval [a, b]. Then there is at least one point c in (a, b) such that
f ' (c) = f(b) - f(a)/ b - a .
A proof would follow if I asked you to demonstrate that the above theorem is true. It would begin with the axiomatic assumptions necesary for the proof, and would follow explicit, logically valid steps using mathematical functions in order to reach the conclusion.
If the proof is valid, then no amount of head-scratching will change that truth value, even if the conclusion is counter-intuitive. That's the case here, I'm afraid. It seems counter-intuitive for anything which begins with 0.x to be able to be the same as 1.0. Counter-intuitive or not, it is true, and it is demonstrably so.
Good luck getting the rust out of those old mathematical skills and studies. I know mine certainly have plenty of rust in them as well, as I don't practice math much these days.
AS
I'm assuming that there is a missing piece in the puzzle for Zep, and once we find what that is, Zep will suddenly "get it". So far, though, we seem to be fumbling around trying different explanations in order to figure out what that piece might be. It's not as if Zep can tell us directly what piece he's missing.
Thank you one and all! Humour noted!
Yes, AS, I do understand about axioms, theorems and proofs. I'm not trying to break the rules here, but use them properly as I go along. Perhaps I built my own stumbling block, as you will see.
All, first off, I'm not trying to invent a solution that arrives at my own pre-specified result here. What I thought I was doing was working from something I know (stuff involving infinite series, in this case - stuff presented earlier) towards something that is fairly new to me (I think what is known as transfinite number theory). By using what I thought were reasonable and logical arguments - extending a series of known properties to infinite terms - I thought I was making reasonable assumptions in doing so. It made, if you will pardon the term, "common sense" to me to do it this way. And in doing so, I ran into some questions about getting from "here" (my own understanding) to "there" (accepted mathematical norms). And these were posed above, if you recall, not as a mechanism to stall but literally as a call for a tutor!
I now realise that my own assumptions are not necessarily acceptable, in that my common sense as I see it is not necessarily mathematical common sense. Or the drink has knocked out too many neural connections...
However I must point out quite clearly that I did understand right from the start about the original proofs provided for the theorem - I was not "missing anything" at all. I thought I said that a few times.
My issue, it appears, has been in making the cognitive leap from what I thought was mathematically logical in my own sense to arrive at the result previously obtained by others. I suppose, AS, that I was finding it hard to accept something that considered "axiomatic" when I could see no reason why it should be so!
I'm still working on this!
Yes, AS, I do understand about axioms, theorems and proofs. I'm not trying to break the rules here, but use them properly as I go along. Perhaps I built my own stumbling block, as you will see.
All, first off, I'm not trying to invent a solution that arrives at my own pre-specified result here. What I thought I was doing was working from something I know (stuff involving infinite series, in this case - stuff presented earlier) towards something that is fairly new to me (I think what is known as transfinite number theory). By using what I thought were reasonable and logical arguments - extending a series of known properties to infinite terms - I thought I was making reasonable assumptions in doing so. It made, if you will pardon the term, "common sense" to me to do it this way. And in doing so, I ran into some questions about getting from "here" (my own understanding) to "there" (accepted mathematical norms). And these were posed above, if you recall, not as a mechanism to stall but literally as a call for a tutor!
I now realise that my own assumptions are not necessarily acceptable, in that my common sense as I see it is not necessarily mathematical common sense. Or the drink has knocked out too many neural connections...
However I must point out quite clearly that I did understand right from the start about the original proofs provided for the theorem - I was not "missing anything" at all. I thought I said that a few times.
My issue, it appears, has been in making the cognitive leap from what I thought was mathematically logical in my own sense to arrive at the result previously obtained by others. I suppose, AS, that I was finding it hard to accept something that considered "axiomatic" when I could see no reason why it should be so!
I'm still working on this!
I was referred to this thread from another thread and I just want to say it is a fantastic thread.
The level of contributions is outstanding and the patience shown astonishing and the willingness to be corrected and to learn an example to all.
It brought a tear to my eye.
The level of contributions is outstanding and the patience shown astonishing and the willingness to be corrected and to learn an example to all.
It brought a tear to my eye.
We're a nice bunch of guys in Science, Mathematics, Medicine, and Technology.Darat said:
The discussion in the puzzles forum is generally even more civil!
It's all those other forums where the fighting takes place. Those religious and philosophical types are real hard cases...