You completely misunderstand the term least upper bound. Why did you subtract 1 in the second to last step?
Friedmanite:
My thoughts exactly. Ok jk, can you explain this? I'm a mere neophyte to mathematicas beyond statistics and college algebra.
One would use the epislon as it means the sum of
No, that's sigma.
However, you are incorrect when you make a distinction between sum and LUB (in this case) when there is none, as Mr. Hardy clearly states. The sum of 1 + 1/2 + 1/4 .... is 2, as Mr. Hardy also clearly states, addressing the very issue presented in the OP.
No. The discussion here is this. Assume Achilles takes a step of length 1/2, then a step of length 1/4, etc. Will he actually ever touch the line in the sand one unit away from where he began. Hardy is going out of his way to say that he never will. 1 is the LUB of his steps. No matter how many steps he takes, even he keeps taking steps forever, he will never ever ever touch the line at distance 1. Now the guys are assuming that he will. They think that the definition of sum of an infinite series [and the corresponding reality] means that at some point he will actually touch the line marked 1.
In any event, go ask your math teacher whether .999... is a number or not. (hint: it's a real number)
Yes, it is a real number, DEFINED [no matter how you construct the real numbers] to be the LUB of the series .9+.09... This LUB happens to be 1. But that is not relevant to our discussion. I'm not denying that the LUB exists.
Sorry you guys are getting hostile. I can do no more but state my position, cite chapter and verse, and refer you to experts. Hey, maybe I'm wrong and you are all right. Sleep well thinking that. It makes no difference to me. last post on this subject.
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It's easy to refute an argument if you first misrepresent it. William Keizer
Jack,
The least upper bound of that series is most definitely 2. Are we understanding least upper bound as the supremum of a bounded subset of an ordered field? Then let,
By definition of the convergence of the partial sums to 2, we have
You cannot add stuff up "ad infinitum".
You are confusing the action of summing with the idea of summation. Any action can only be performed finitely many times. But an idea - such as a point, line, number or plane - can effortlessly be conceived to be in infinite quantity. The idea of summation is no different than the idea of a point, line or plane and no less distinct from the act of summing than the idea of a point is from drawing a dot on paper.
Conceiving of an infinitude of sums is no more difficult than conceiving of an infinitude of numbers. For example, just as we can construct the natural numbers N, we can also construct a similar set of sums N+ such that for each element n of N, the corresponding sum in N+ is n+1. This technique is used, in fact, in the very definition of infinity. That an infinitude of sums should all contribute to the same final sum is no more difficult to conceive than N+.
The issue here is metaphysical, not mathematical. If you banish actual infinity from your mathematics, you are left with this timid jot-tittling about least-upper bounds. If you banish all talk of "potential infinity" and embrace actual infinity, you get uncluttered mathematics.
Clayton -
You are confusing the action of summing with the idea of summation.
No. I'm talking about the idea of summation. As is Hardy, and every other math book you will ever open.
But an idea - such as a point, line, number or plane - can effortlessly be conceived to be in infinite quantity.
The problem, however, is defining things. Lines and planes are just sets satisfying ceratin conditions. But "adding infinitely many things together" needs a definition, which you are not providing. You see, addition is defined for two objects intially, and for any finite number recursively. But no definition has yet been found for adding infinitely many things other than defining "the infinte sum" to be one thing, and one thing only. The LUB [or unique accumalation point, if there are negative terms].
Conceiving of an infinitude of sums is no more difficult than conceiving of an infinitude of numbers.
No. It is much more difficult. Because it has to be defined. An infinitude of numbers has a clear definition. A set S has an infinitude of elements if it can be placed in one to one corrrespondence with a proper subset of itself. [The link you provided to the Axiom of infinity is not the same as the definition of infinity. "Axiom of infinity" is just an informal name given to that axiom.] What is your definition of an infinite sum? Lay it out so we can see.
The definition of N+ you provide only works for a finite number, but not for an infinite number. And if you think it is as easy to concieve of an infinite sum as to conceive of N+, please provide a definition.
Nobody has banished actual infinity [=transfinite sets] from math, even those who pay attention to detail and insist on least upper bounds.
It surprises me that you dismiss least upper bounds as jot-tittling. Do you understand why the math community started talking about least upper bounds? What difficulties were they facing that made them resort to such a concept? Was it love of jot tittling for its own sake, do you think?
You cannot banish all talk of potential infinity, because it is a solid irrefutable concept. Aristotle nailed it. Nobody disagrees with him.
Non standard analysis has nothing to do with the discussion at hand.
Friedmanite,
The LUB of 1 + 1/2 + 1/4 + ... is 2. The LUB of 1/2 + 1/4 +... is 1 one less than that.
1/2
3/4
7/8
15/16
31/32
63/64
127/128
255/256
511/512
1023/1024
2047/2048
......
When does the sum become greater than 1?
faber est suae quisque fortunae
You seem to be laboring under the delusion that mathematics is a monolithic subject like logic or factual truth - it is not. You can build fundamentally incompatible - yet internally consistent - mathematical formalisms by choosing different sets of axioms. So, your claims about "all mathematicians" or "every math book" strike me as hyperbolic and dogmatic.
The problem, however, is defining things. Lines and planes are just sets satisfying ceratin conditions.
Sets of what? Points? Numbers? What is a point? What is a number? What is a sum? Why can you conceive an infinitude of points but not an infinitude of sums?
But "adding infinitely many things together" needs a definition, which you are not providing.
Well, I haven't yet, but I provided N+ as a preliminary step to this.
Before we get there, let's start with the infinitude of sums, N+.
OK but the point is that we can conceive of an infinitude of sums. The next step is to arrange them in such a way that they are all connected to one another. For this, the lowly Fibonacci series will work quite handily:
f_0 = 1 f_1 = 1 f_n = f_n-1 + f_n-2, for all n>1
The set of all terms in the Fibonacci series can be proved to exist.
F = { f_n }
F is an infinite set. Yet every member of F is connected to its neighbors in a sum (recursion). Even though it diverges, it is formally an infinite sum.
Ironically, this is similar to the recursive construction of N in Peano arithmetic, cf the sixth axiom listed here.
The final question is whether such formal, infinite sums can ever converge to a finite sum and the answer is... yes! And this is the answer to Zeno's paradox.
Nobody has banished actual infinity [=transfinite sets] from math, even those who pay attention to detail and insist on least upper bounds. It surprises me that you dismiss least upper bounds as jot-tittling. Do you understand why the math community started talking about least upper bounds? What difficulties were they facing that made them resort to such a concept? Was it love of jot tittling for its own sake, do you think?
Well, least upper bounds and other limiting concepts are required if you subject yourself to the discipline of developing analysis while denying the existence of actual infinities.
Not true at all. Potential infinity is a denial of the existence or possibility of actual or completed infinity. The two concepts are incompatible. Either you accept completed infinities (transfinite magnitudes) or you do not. Even Hardy in the first Appendix briefly discusses actual infinity in connection with projective geometry. He admits its existence in a different area of mathematics but refuses to admit its use in analysis. That's his choice but I don't see what is gained.
No contradictions arise from the use of actual infinities in the development of analysis so long as it is understood that no transfinite magnitude is a member of the set of numbers being studied (natural, rational, real, complex, etc.)
Sure it does - non-standard analysis incorporates infinity (and 1/infinity) as elements of the hyper-real numbers. All the usual theorems of analysis are developed with ease.
Clayton,
I'm glad to see you are reseraching this fascinating topic.
Unfortunately, you won't be able to get why everything you wrote is either wrong or irrelevant from my reply to you. Get thee to a human Math Professor in your local area.
My last post [again!].
I have to say, everyone has given a valiant effort at this thread. Thanks to all for a great discussion.
Friedmanite: Jack, The least upper bound of that series is most definitely 2. Are we understanding least upper bound as the supremum of a bounded subset of an ordered field? Then let, By definition of the convergence of the partial sums to 2, we have
Saw a video on youtube and noticed that this guy didn't get anywhere close to answering the Achilles vs. Tortoise question.
I still maintain that it's because numbers, time and infinitesimal quantities don't exist in the first place.
Since the nineteenth century, infinitesimal quantities have been eliminated from math [but they can be re-introduced rigorously if we want to, with non standard analysis].
As for numbers and time not existing, surely the question is why are the usual assumptions made with time and numbers adequate to explain everything else, but fall apart at Zeno's paradox?
I agree that the video doesn't explain anything. Nobody ever doubted that the sum is less than a minute. The question is how do we ever get to a minute, if we are always less than it?
Holy cow what a huge thread! I must not have been an active member when this was out. But the way I approach the paradox (just read it), I'd say empirical evidence shows that Achilles will certainly beat the tortoise, so the rest doesn't matter. Mathematically, maybe we're not intelligent enough to figure it out, idk. But it's like wondering about how the universe started. Just because we don't know how is could ever start doesn't mean we aren't here. We are here, so theories just need to catch up. That's my initial take on it.
The part about math not existing, or numbers, however it was put, does make a great deal of sense when you think about it. Then again I'm not a mathematician, so I wouldn't want to be hasty and blindly accept that notion right off.
I'm going by Wikipedia's summary of the paradox. In short, the tortoise starts 100 meters ahead of Achilles. After they start, Achilles will reach the tortoise's starting point, at which point the tortoise will be at some new point farther along. Achilles must then reach this point, and we're back at the same essential situation. Repeat indefinitely; Achilles will always be reaching points the tortoise was previously at, and never catch up. My resolution is that we might imagine that each iteration of the above is forward in time by a roughly equal amount, but it's not; each iteration is going forward a smaller and smaller amount of time, such that we always "slow time down" enough that we never get to the moment he actually does pass the tortoise. When he's almost there, our next step forward in time is extremely small. He's now closer, but we'll then step an even smaller amount forward in time. It's like taking a gold coin and dividing it in two and discarding one of the halves, then dividing the remaining halve in two, indefinitely; we're always removing gold, yet we'll never run out, since we're discarding less and less on each iteration.