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19th January 2019, 02:22 PM  #3281 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,872

Wholeness is not necessarily Comleteness
Wholeness is not necessarily Completeness, as seen in http://www.internationalskeptics.com...postcount=2798 exactly because infinitely many things are infinitely weaker that actual infinity (as seen in http://www.internationalskeptics.com...postcount=3095).
In order to deal with such notions, philosophy and mathematics are inseparable of each other (http://www.internationalskeptics.com...postcount=3280). 
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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 January 2019, 04:36 AM  #3282 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,872

Please look at the following diagram:
It was known as "2X=X√2 paradox" (This is an old "problem" that was known at least to Leibniz and probably to the Greeks). Actually, this is not a paradox at all since no integer is an irrational number, and a straightforward way to show it, is by X=1, that is, 2>√2. By observing the top of the attached diagram, one finds the convergent series a+b+c+d+... 1) Please pay attention that this series is rigorously defined by the intersections of the black straight lines (which go through the peaks of the zigzag (black, red, green, magenta, blue, cyan) lines with constant length 2X) with each side of the square. 2) It means that the mathematical fact that 2X>X√2, is inseparable of the mathematical fact that 2X>2(a+b+c+d+...). Let X (one side of the square) = 1 In that case (a+b+c+d+...) is actually (1/2+1/4+1/8+1/16...). By (2) 2(1)>2(1/2+1/4+1/8+1/16...), which can be reduced into 1>1/2+1/4+1/8+1/16... It has to be stressed that no partial sums like a, a+b, a+b+c, ... are involved in this argument, but not less than the series a+b+c+d+... If one does not agree with the argument above, one has to prove (according to the considered diagram) that series a+b+c+d+... is not defined by the zigzag lines (where, again, no partial sums like a, a+b, a+b+c, ... are involved in such proof). Moreover, if one proves it, one also demonstrates why visualization is insufficient for rigorous mathematical results. I am fully aware that what is called "not a summation in the usual sense" means a+b+c+d+... ≤ X, where the semantics (meaning) of ≤ (in the considered case) is "not greater than" X, or "at most" X. Since series a+b+c+d+... is strictly defined by all the zigzag lines such that 2X is strictly > X√2, series a+b+c+d+... can't be but strictly < X. So I still do not see how ≤ is relevant to the diagram above. 
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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 January 2019, 07:21 AM  #3283 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,872

I wish to stress that, for example:
S = 1/2+1/4+1/8+1/16... 2S = 1+1/2+1/4+1/8+1/16... 2S  S = 1  S is not a proof of the considered case because: 1) By omitting S from 2S there is no guarantee that the omitted value (= 1/2+1/4+1/8+1/16...) is equal to the nonomitted value (= 1). 2) The separability between 2>√2 and 1>1/2+1/4+1/8+1/16... has not been proven. 
__________________
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 January 2019, 07:30 PM  #3284 
Graduate Poster
Join Date: Nov 2006
Posts: 1,917

error

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I'm an "intellectual giant, with access to wilkipedia [sic]" "I believe in some ways; communicating with afterlife is easier than communicating with me." Tim4848 who said he would no longer post here, twice in fact, but he did. 

31st January 2019, 08:30 AM  #3285 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,872

The standard notion of set (according to Prof. Melvin Randall Holmes):
Quote:
By logically going beyond the notion of collection {} is tautology and {} is contradiction, such that any given collection is ~contradiction AND ~tautology. As about cardinality: {} = 0 {} = ∞ = the cardinality of actual infinity {...} = any cardinality > 0 AND < ∞ Some examples: {{}} = 1 {{}} = ∞ {{}} = 0 {{1,2}} = 1 {{1,2}} = 2 {1,2} = 2 {{1,{},2}} = 0 {{1,{},2}} = ∞ Nested cardinality examples: {{1,{},2}} = (((0)3)1)∞ {{1,{},2}} = (((0)1)1)∞ {{1,{},2}} = (((0)1,1)1)∞ {{1,{},2}} = (((0)1,1,1)1)∞ {} = (0)∞ etc. ...  As can be seen, the standard notion of collection is a very limited mathematical framework. 
__________________
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 ) 

2nd February 2019, 11:52 PM  #3286 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,872

By going beyond the notion of collection (which is a composed thing) the noncomposed is defined by noncomposed opposite extremes, which are NOthing and YESthing, where the cardinalities (the magnitudes) of them are {} = (0)∞
So the cardinality of any give collection is > 0 AND < ∞, which means that no collection is accessible to that has cardinality 0 (NOthing) or cardinality ∞ (YESthing). By being aware of the composed and the noncomposed, one enables to understand why a collection with endless members is not actual infinity, simply because it is inaccessible to YESthing (that has cardinality ∞). 
__________________
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 ) 

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