There is no answer to my question.
Deeper than primes
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Cantor proved that the assumption of every point on the real line being accounted for is false. The "smallest line segment" does exist but it is undetectable as much as the number of points on the real line is uncountable. You can somewhat get the notion of the uncovered space by looking at irrational numbers and some rational numbers approaching their limits, such as 3.1415... approaching its limit Pi, or 0.1111... approaching its limit 1/9. Both limit points Pi and 1/9 do not really exist, coz they are unreachable within the type of real line used by Cantor for his diagonal prove. You can see that the diagonal method involves the approximate (radix) format, as it must.I can't show you the smallest line segment, because it does not exist, which proves that a line is completely covered by points. How about I show you the dumbest a human can get?
Since the value of an irrational number infinitely increases in a convergent manner, it cannot reach a certain value, which is called "the limit." So there must be always some space left. This is all irrelevant to geometry, coz locating points with infinite precision is good enough for any sane person.
How can a smallest line segment exist if for every line segment you show there is a smaller one?
There is no smallest line segment as much as there is no "biggest number." There is a space opening when limits are present
[lim n → ∞] 1/n = 0
The above tells you that when n grows unbound like 1, 2, 3, 4, . . . , the length 1/n of each segment of the line being divided ad infinitum is approaching zero, but it can't ever reach it, coz n grows without bound. The intuition tells you that if you try to reach something and can't, there is always some space between your hand and the object. And that's what Doron has been looking at.
The notion that the line is entirely covered by points is a part of the definition of the line -- it's not a theorem to be proven -- and the definition has worked fine for the purposes of Euclidean geometry. But there are different conditions in the set theory where the set of real numbers can be also likened to points on the line. Cantor's set theory doesn't just treat infinity as geometry does -- it attempts to compare one infinity to another infinity, like the infinitude of natural numbers to the infinitude of rational numbers, so some implied "spaces" between these numbers should be taken into account. Here, the intuitive thinking may fail, as it did when Cantor published his results to shake up the intuitive understanding of infinity that happened to be correct just for the practical purposes of computing. Facing the requirements of the set theory, the full point coverage idea must be carefully defined with non-verbal terms. I can't do that though.
Hit a nerve have I? The fact that I learned some math for my degree has nothing to do with arrogance. Humbleness allows learning from others - a quality you are lacking.
What on earth does any of this topic have to do with the "fringes of atheism", whatever they are?
doronshadmi
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No, you are missing the nerve, actually your own nerve, because instead of using your own reasoning in order to really examine what enables the existence of different smallest elements along a line segment (such that no sub-line segment has the form of the smallest element), you let your "humbleness" to continue the ignorance that you are learning from others.
In other words, you prefer to be one of the herb instead of really get to the "heart and bones" of this non-trivial fine subject.
Please show us the smallest line segment by using your "humbleness" in order to use the reasoning that you are learning from others.
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doronshadmi
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In other words, you do not understand (yet) the following:
http://www.mathacademy.com/pr/prime/articles/cantor_theorem/index.asp is a clear example of Cantor's theorem as a proof by contradiction, which leads to contradiction if one tries to define mapping between an explicit P(S) member and S member, because of the construction rules of the explicit P(S) member (the member of S must be AND can't be a member of the explicit P(S) member, according to the construction rules of the explicit P(S) member, under Cantor's theorem).
Also since |P(S)| is not less than |S| (because it is trivial to show that all S members (for example {a,b,c,d,...}) are at least mapped with {{a},{b},{c},{d},...} P(S) members), then by using this fact and the contradiction shown above, one must conclude that P(S) is a larger set than S.
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But this is not the only way to look at this case, for example, we are using Cantor's construction method to systematically and explicitly define P(S) members, for example:
By using the trivial mapping between {a,b,c,d,...} S members and {{a},{b},{c},{d},...} P(S) members, we explicitly define P(S) member {}.
Also by using the mapping between {a,b,c,d,...} S members and {{},{a},{b},{c},...} P(S) members, we explicitly define P(S) member {a,b,c,d,...}.
Actually by using Cantor's construction method independently of Cantor's theorem, we are able to explicitly define the all P(S) members between {} and {a,b,c,d,...}.
As a result, there is a bijection between S and P(S) members, as follows:
a ↔ {}
b ↔ {a,b,c,d,...}
c ↔ some explicit P(S) member, which is different than the previous mapped P(S) members
...
etc. ... ad infinitum.
Please be aware of the fact that this construction method has nothing to do with Cantor's theorem exactly because the construction is used independently of Cantor's theorem and therefore it is not restricted to the logical terms of Cantor's theorem.
In other words, the distinction between countable and uncountable infinite sets has no basis.
Actually Dedekind's definition for infinite sets (which according to it there is a bisection between any given infinite set and some proper subset of it) is true also in the case of the mapping between the members of P(S) and the members of S, where S is indeed a proper subset of P(S).
The bijection between the members of P(S) and the members of S is equivalent to the bejection between the members of N and the members of even numbers, or the members of Q and the members of N, etc ...
Yet no one of these sets is complete, because no infinite set of distinct objects has the magnitude of Fullness, which is that has no successor (and a line is the minimal expression of Fullness).
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OK, I'm glad you agree.
What are you talking about? Cantor's proof of R being an uncountable infinite set got nothing to do with Cantor's theorem, which deals with a size relation between a set and its power set and leads toward the proof that the power set of any countably infinite set is uncountably infinite. I was explicitly referring to the proof of uncountability of R.
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
Have you missed the large numbers picture with 3.14159... on top in the relevant post?
doronshadmi
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You are wrong.
א (which is the cardinality of R) and 2א0 (which is the cardinality of P(N)) have the same magnitude.
doronshadmi
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Are you kidding?
I am the one who argue all along this thread (which does not agree with the traditional argument about this subject) that 3.14159...[base 10] < pi
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Math operations are mostly done with numbers in exact format, that means with the limits. For the practical use, the result of the operation in the exact format has to be converted into the aproximate format, coz Log(25) + Cos(pi/3) pounds is not what you find on the weight scale. Pi also needs to be converted into the usable approximate format, so this identity appears: Pi = 3.14159... But that's not true, coz "pi" is a symbol for the limit whose real value is uknown and which exists only in the exact format, such as pi=circumference/diameter. The real value becomes more and more precise ad infinitum, and obviously converges. Since there is no correspondence between infinity and physical realities the fact that 3.14159... doesn't equal pi but is getting infinitely close to it is infinitely negligible.
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I'm not sure I see the relevance of mentioning the base. I also don't think you can show that anyone in this thread has said that if you stop the evaluation of pi at any arbitrary number of places that it isn't less than the full value. However, since you have used ellipses, that implies an infinite expansion, which would be equal to pi. Again, your limitations trip you up.
You think you are smarter than everyone else, yet call others arrogant. Another contradiction?
doronshadmi
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zooterkin, pi is an exact location along the real line, so there is no evaluation of pi.
On the contrary 3.14159...[base 10] is not pi exactly because any given smaller sub-line segment is irreducible into the smallest element, which is the point at the exact location of pi, along the real line.
doronshadmi
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You still do not get that the answer to this non-trivial subject is not given by any one but you, sympathic.
Instead of doing the must have journey into your own mind in order to really deal with the considered fine subject, you are using only the reasoning that you are learning from others.
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doronshadmi
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The only journey you need to take to understand these elementry concepts is to your local college math classroom.
doronshadmi
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You did this journey, so please use it in order to define the smallest sub-line segment.
Be aware of the fact that no sub-line segment is reducible to the smallest existing element, which is a point.
If you can't define the smallest sub-line segment, you have no choice but to conclude that no collection of the smallest elements (known as point) completely covers an element that it is irreducible into a point.
Actually without the existence of the ever smaller elements between any arbitrary closer points, there are no distinct points at all.
Furthermore, the journey you made can't help you to get http://www.internationalskeptics.com/forums/showpost.php?p=7066349&postcount=14908 .
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