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27th April 2015, 07:48 AM  #321 
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Please follow the thread by starting from http://www.internationalskeptics.com...&postcount=314.
Thank you. 
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27th April 2015, 08:26 AM  #322 
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27th April 2015, 09:02 AM  #323 
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The question is at the end of http://www.internationalskeptics.com...&postcount=314 and it is an inseparable part of this post.

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27th April 2015, 09:05 AM  #324 
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27th April 2015, 11:02 AM  #325 
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By following after the standard definition (as seen in http://www.internationalskeptics.com...&postcount=314), what is your answer to this question?

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27th April 2015, 11:13 AM  #326 
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By following after? The standard definitions are the only ones that matter. Any other meaning you might imagine for, say, limit wouldn't be limit any more. It would be something else. You cannot show Mathematics to be broken simply by changing the meaning of things.
Quote:

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27th April 2015, 11:23 AM  #327 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

27th April 2015, 11:42 AM  #328 
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27th April 2015, 11:59 AM  #329 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

27th April 2015, 12:06 PM  #330 
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27th April 2015, 12:17 PM  #331 
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∞ does not rigorously define infinity, so since "Infinite geometric series" and "Proof of convergence" use it, they do not define any rigorous proof.
A rigorous proof must be based on infinity as defined by transfinite cardinality for example: aleph0 < aleph1 < aleph2 < ... etc. 
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27th April 2015, 12:24 PM  #332 
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Oh, I see. It is limits you don't understand. Odd that, since that is the very thing you have been trying to attack. Good luck with that. It is usually best to understand what you object to rather than simply object. I take that back. It is not just limits you don't understand for the matter at hand. Infinite sequences escape your grasp, too. Otherwise, you'd know why your statement quoted here is, as they would say on the old Perry Mason TV series, incompetent, irrelevant, and immaterial. 
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27th April 2015, 12:37 PM  #333 
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It does not change the fact that a rigorous proof must be based on infinity as defined by transfinite cardinality for example: aleph0 < aleph1 < aleph2 < ... etc.
∞ does not provide the necessary rigorous terms exactly because it does not distinguish between aleph0, aleph1, aleph2, ... etc. 
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27th April 2015, 12:45 PM  #334 
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As I said, your failure to understand limits and sequences allows you to believe your objection has substance. It does not. Why it does not is basic to both concepts.
However, just so you don't get completely hung up on a pro forma objection, one without substance, perhaps you might consider the question of where would the proof break down if your objection did have substance. Put another way, what purpose does the infinity used in the proof serve, since at no point does infinity itself in any of its possible forms ever actually enter into the argument. 
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27th April 2015, 06:34 PM  #335 
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27th April 2015, 06:56 PM  #336 
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27th April 2015, 07:35 PM  #337 
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28th April 2015, 02:11 AM  #338 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

28th April 2015, 02:17 AM  #339 
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There is no rigorous proof by using inaccurate form like ∞ for infinity.
Infinity itself (notated by ∞ inaccurate form) actually enters into the argument, exactly as seen in "Infinite geometric series" section of Geometric progression^{WP} or in "Proof of convergence" section of Geometric series^{WP}. 
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28th April 2015, 04:49 AM  #340 
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Good thing, then, the proof uses no form of infinity.
Quote:
If you think otherwise, go ahead and identify exactly where "infinity" enters into the prove (as something other than mere notation for 'limit'). 
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28th April 2015, 05:09 AM  #341 
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Dear jsfisher are there finitely or infinitely many elements in sequence (0.9_{10}, 0.09_{10}, 0.009_{10}, ...) that all of them are used in order to construct 0.9_{10}+0.09_{10}+0.009_{10}+ ... = 0.999..._{10} (such that 0.999..._{10} is not less than the sum over all (0.9_{10}, 0.09_{10}, 0.009_{10}, ...) elements) ?
 Once again: for all ε > 0, there exists a natural number N in N such that for all n ≥ N, a_{n}  L < ε. So a_{n}  L must be = 0 in order to conclude that the sum over a_{n} = L. It must be stressed that if infinity is not involved here, then a_{n}  L > 0 and we can't conclude that the sum over a_{n} = L. For example, where is the rigorous proof that the sum over all (0.9_{10}, 0.09_{10}, 0.009_{10},...) = L = 1? (My solution is given in http://www.internationalskeptics.com...&postcount=316). 
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28th April 2015, 06:57 AM  #342 
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Nowhere does that enter into the proof. Why to you harp on the irrelevant?
Quote:
Quote:
Quote:

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28th April 2015, 11:25 AM  #343 
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In that case it is perfectly acceptable that 0.999..._{10} < 1 by a_{n}  L > 0, or in other words
Quote:
Moreover, you actually ignore this (http://en.wikipedia.org/wiki/Sequenc...and_infinite):
Quote:

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28th April 2015, 12:16 PM  #344 
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The two are unrelated. My observation is supported by the very definition of limit. Yours is not supported by anything.
Quote:
Quote:
Quote:
Determining the limit of the sequence of partial sums does not involve infinity at any step. 
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28th April 2015, 08:40 PM  #345 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

29th April 2015, 12:10 AM  #346 
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Exactly as given in http://www.internationalskeptics.com...&postcount=316.

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29th April 2015, 12:53 AM  #347 
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The very definition of limit, if it is limited to finite sequences, does not support the assertion that the sum over all terms of sequence (0.9_{10}, 0.09_{10}, 0.009_{10}) actually = 1.
Moreover, the sum of N terms of that sequence < 1 , and we need a gap closer at R size in order to actually define 1 as the limit of 0.999..._{10}. 0.999..._{10} is not identical to 1, and the sequence of partial sums, even if it has N terms, does not establish  rigorously  that 0.999..._{10} is identical to 1, simply because without using alsoR size, 0.999..._{10} < 1 by exactly 0.000...1_{10}, where 0.000... is a place value keeper that the number of its places is in bijection with the number of places in 0.999... . The number of places is at most at N size, so in order to actually close the gap between 0.999..._{10} and 1, ...1_{10} that is at R size is needed to actually do the job. In other words: 0.999..._{10} + 0.000...1_{10} = 1. More generally, the sum over a_{n} where a_{n} has at most N terms < L by sum at R size. 
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29th April 2015, 02:56 AM  #348 
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Let's clarify it by a given example that can be used without loss of generality:
The sum over (0.9_{10}, 0.09_{10}, 0.009_{10}, 0.001_{10}) = 1 The sum over (0.9_{10}, 0.09_{10}, 0.009_{10}, 0.0009_{10}) < 1 Please pay attention that in this finite example the sum = 1 if we are using the smallest value 0.001_{10} that is at higher level than value 0.0009_{10}. In the case of the sum over infinity many values like (0.9_{10}, 0.09_{10}, 0.009_{10}, 0.0009_{10}, ...) the smallest value > 0 (notated as 0.000...1_{10}) is determined as the sum at R size (notates here as ...1_{10}), which is at higher level than the sum at N size (notate here as 0.000..., and 0.000...1_{10} is taken as a unique value exactly as 0.001_{10} is taken as a unique value. 
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29th April 2015, 05:41 AM  #349 
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According to the example above, one can understand that a sum over sequence with R terms (which can't be symbolized since it is uncountable) needs ...1 at P(R) size in order to be identical to 1, ... etc. ad infinitum, so number systems (whether they are countable or uncountable) are determined by different levels of resolutions, where the higher resolution is taken as one unique value (notated as ...1) from the point of view of the lower resolution, and it closes the gap between the limit value and the sum of the lower resolution.
On top of all this there is the noncomposed highest level that is beyond all resolutions, and it closes the gap to 1 for all resolutions. In other words, the noncomposed highest level is independent of any point of view, in order to be used to close the gap between the sum of a given resolution, and 1. So, that is beyond transfinite cardinality is indeed actual infinity, and it is invariant and objective to all levels of collections, that each one of them or all of them together are not invariant and objective as the noncomposed level is.  If we use the point of view of Physics on this case, then Newtonian physics is useful even if it is not accurate as GRT, or in other words, by using this analogy, the relative different points of view of different resolutions are still useful in order to close the gap between some sum at smaller transfinite cardinality to some limit value, by using a given sum of bigger transfinite cardinality (which is equivalent here to Newtonian physics). Yet the total solution is provided only by using the noncomposed highest level that is beyond all resolutions (which is equivalent here to GRT, in this analogy).  Definitions like "for all ε > 0, there exists a natural number N in N such that for all n ≥ N, a_{n}  L < ε", simply block the ability to understand the relative view of different levels of cardinality (or the concept of relative levels, in general), and the absolute view of the noncomposed level beyond all transfinite cardinalities, and how the relative or the absolute views are used in order to solve the limit problem. 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

29th April 2015, 07:27 AM  #350 
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Let's not. You continue to show complete disregard for and lack of understanding of how things are defined. Despite your objections, the series for an infinite sequence is, by definition the limit of the sequence of partial sums. Determination of that limit does not at any point involve infinity, in any of its possible forms.
If you are unwilling to accept and comply with fundamental mathematical definition, then you are not doing Mathematics. You, doronshadmi, are not doing Mathematics. 
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29th April 2015, 07:43 AM  #351 
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You, jsfisher, put arbitrary restrictions to mathematical development, and use them in a dogmatic way.
If you are unwilling to reexamine fundamental mathematical definitions, then you are not doing Mathematics. This time please reply to http://www.internationalskeptics.com...&postcount=345. 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

29th April 2015, 07:53 AM  #352 
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As you have been invited many times in the past, if you want to explore new areas of Mathematics, do so. However, you are not allowed to redefine established terminology and declare prior results wrong as a consequence.
Quote:

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29th April 2015, 08:10 AM  #353 
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I can reexamine or redefine your so called "established terminology", and this is a perfectly valid way of mathematical development.
The difference between you and me is that since you are now a moderator, you can shut me up because I do not follow after your so called "established terminology". But before you going to shut me up, be aware of the fact that this thread is a part of "Religion and Philosophy" forum, which somehow you became one of it moderators. 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

29th April 2015, 08:13 AM  #354 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

29th April 2015, 08:25 AM  #355 
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29th April 2015, 09:42 AM  #356 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

29th April 2015, 10:39 AM  #357 
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29th April 2015, 11:28 AM  #358 
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I am very specific in:
http://www.internationalskeptics.com...&postcount=347 http://www.internationalskeptics.com...&postcount=348 http://www.internationalskeptics.com...&postcount=349 where limit is not restricted only to finitism, and relative or absolute observations are used. In all cases (depending on relative or absolute observations) a sum over a given sequence reaches the limit (and not just tends to, as defined by your finiteonly observation). 
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As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

29th April 2015, 12:22 PM  #359 
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29th April 2015, 09:02 PM  #360 
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