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Deeper than primes

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jsfisher said:
Are you perhaps now claiming you did provide a bijection between the members of {} and {{}}??
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http://www.internationalskeptics.com/forums/showpost.php?p=7019615&postcount=14694 is not about a bijection, but it is about fundamental notions about the concept of Set.

By Traditional Set theory, Nothing is not considered as a mathematical thing, so in this case we get:

{} ↔

where {} is the member of the powerset of {}.


Since Nothing is not considered as a mathematical thing (by Traditional Set theory), then the minimal existing set is the Empty set and by following the restriction of "Set is an existing mathematical object", we get the following bijection, which is between the Empty set and the member of the powerset of the Empty set (which is actually the Empty set), as follows:

{} ↔ {}


jsfisher's weak reasoning simply can't comprehend what is written above.
 
epix said:
The only thing that can be understood from your reference is that involves infantile trivialization of the subject, e.g. after mummy comes home, daddy comes home.
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Wrong, it is a simplification of a mambo jambo complication.
 
jsfisher said:
Doron, stop trying to change the subject. You made a claim. You were called out on it. You have tried repeatedly to support the claim, but every single time you tried to support it with an example you have failed.

You failed every single time.
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That reminds me that epix who have been trying to disect the set of all atheists, but epicly failed each time he tried. LOL.
 
doronshadmi said:
Since the Circle's game can't be comprehended by you, I have changed it to the Line segment game, as follows:

Any additional point between the extreme endpoints of a given line segment, is resulted by more lines with end points, etc. ad infinitum, where the points are actually different than each other exactly because given any scale level, there is an uncovered line between any closer pair of points (the closest pair of points does not exist exactly because there is always an uncovered line between them) along the finitely long line.
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Back to this one again? What uncovered line? Please show where there is a part of the line which has no point on it.
 
doronshadmi said:
http://www.internationalskeptics.com/forums/showpost.php?p=7019615&postcount=14694 is not about a bijection, but it is about fundamental notions about the concept of Set.
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Yes, that was obvious to everyone including, apparently, even you.

However, the thing you were trying to present has supposed to be a bijection, remember? You were to provide a bijection, any bijection at all, between the members of {} and {{}}.

Try as you may, you keep failing at showing one. You did claim one existed, yet, you cannot point one out. Instead, you keep trying to change the subject.
 
doronshadmi said:
Wrong The Man,

The set of all circles with different curvatures is associated with all possible points along the infinitely long straight line, and yet there is discontinuity between these associated points and the center point, and the infinitely long straight line is permanently beyond the set of these associated points, because of a very simple reason:
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The discontinuity is again only in your head Doron. Your center point is a point on your line as specifically designated by you. You remain with no part of your line that is not covered by points even in your “discontinuity” with yourself.


doronshadmi said:
pi is not a property of the center point (total curvature) or the infinitely long straight line (total non-curvature).
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Points still aren’t circles, no matter how much you would like them to be or insist that something is lacking because they aren’t.

doronshadmi said:
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Since the Circle's game can't be comprehended by you, I have changed it to the Line segment game, as follows:

Any additional point between the extreme endpoints of a given line segment, is resulted by more lines with end points, etc. ad infinitum, where the points are actually different than each other exactly because given any scale level, there is an uncovered line between any closer pair of points (the closest pair of points does not exist exactly because there is always an uncovered line between them) along the finitely long line.
Click to expand...

Your game still remains the same; you are just fixated on the smallest and the largest whether they are circles or line segments. You simply insist that something is lacking if they are indefinable in some set, yet fail to understand (again apparently deliberately since it has been explained to you many times) it is exactly that lacking of a definable smallest and/or largest circle and/or line segment which guarantees that your entire line is covered by points (a continuum) as well as line segments and even concentric circles (when not centered on your line).
 
The Man said:
Points still aren’t circles, no matter how much you would like them to be or insist that something is lacking because they aren’t.
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But they are associated with the set of all circles with different curvatures, where these associations can't completely cover an infinitely long straight line, which is intersected by all the circles of this set.

The Man said:
it is exactly that lacking of a definable smallest and/or largest circle and/or line segment which guarantees that your entire line is covered by points
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Exactly the opposite.

For example let us focused on the line segment case:

The existence of any arbitrary pair of points along a line segment is guaranteed by the existence of an uncovered line between them.

Without this uncovered line, the pair is merged into a single point.
 
The Man said:
Points still aren’t circles, no matter how much you would like them to be or insist that something is lacking because they aren’t.
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Doron sees as the circle is getting smaller and smaller in one instance resembling just a dot. It doesn't take much in his case to switch from 1-dim object to 0-dim object and treat the tiny circle as a point. Doronian equalities such as 1=0 are the banners of his crusades under which he keeps invading the Land of Reason hoping to subdue it. You can see another "equality" as Doron tries to outflank jsfisher's position near the power set. According to Doron, any finite set has the same cardinality as the power set, e.g. n = 2n. These "equalities" are the result of The Super Axiom of Omnipotence, the sole axiom that OM is built upon and which states that 1 + 1 = 3.

Note: In the opposite case, when the circle is getting larger and larger, its circumference gets mercifully out of Doron's sight, and so he can't transform it into some other object like in the case of his jolly circle/point transformation.
 
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epix said:
Doron sees as the circle is getting smaller and smaller in one instance resembling just a dot. It doesn't take much in his case to switch from 1-dim object to 0-dim object and treat the tiny circle as a point. Doronian equalities such as 1=0 are the banners of his crusades under which he keeps invading the Land of Reason hoping to subdue it.
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"Brilliant" conclusion epix. I clearly write that a circle or a line can't be a point (and this is a exactly the reason of why the smallest line or circle do not exist), but you, by using your twisted reasoning, get exactly the opposite (1=0).


epix said:
You can see another "equality" as Doron tries to outflank jsfisher's position near the power set. According to Doron, any finite set has the same cardinality as the power set,
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Wrong, by using Cantor's construction method, one enables to explicitly construct the all possible members of P(S), and because of this fact, one enables to define mapping between these explicit members and the same amount of members, taken to from the set of natural numbers, or any other set with different objects.

Furthermore, the mapping exists between no mapping with P(S) members, and a 1-to-1 and onto with P(S) members, and the degree of mapping is chosen according to one's needs (there is no universality about mapping).

epix said:
Note: In the opposite case, when the circle is getting larger and larger, its circumference gets mercifully out of Doron's sight, and so he can't transform it into some other object like in the case of his jolly circle/point transformation.
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Wrong again epix, since all circles have some curvature > 0 AND < ∞ , then:

1) The largest circle does not exist and as a result, the set of all circles can't completely cover an infinitely long straight line, exactly because such an object has exactly 0 curvature, and as a result it is permanently beyond the range of the set of all circles with different curvatures.

2) The smallest circle does not exist and as a result, the set of all circles can't completely cover an infinitely long straight line, exactly because such a collection do not have an object with ∞ curvature, which is exactly a point (the center point is permanently beyond the range of the set of all circles with different curvatures, which are obviously < ∞ curvature).
 
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doronshadmi said:
Good.

In that case there is an uncovered line between any arbitrary closer pair of points, along any given line.
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:rolleyes:

What makes you think that segment is 'uncovered', as you call it? The points exist, they are there, you do not need to place them to bring them into existence. There is nowhere on the line, whether between two other arbitrary points or not, that does not have a point on it.
 
doronshadmi said:
Good.

In that case there is an uncovered line between any arbitrary closer pair of points, along any given line.
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In addition to the insertion of the word, uncovered, that makes your statement completely bogus, this is another example of your sloppy powers of expression, Doron.

You didn't not mean "between any arbitrary closer pair of points". You meant that there must exist some pair of points with an "uncovered" line segment between them.

The statement is still wrong, as zooterkin points out. It is just less wrong this way.
 
zooterkin said:
:rolleyes:

What makes you think that segment is 'uncovered', as you call it? The points exist, they are there, you do not need to place them to bring them into existence. There is nowhere on the line, whether between two other arbitrary points or not, that does not have a point on it.
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Without the uncovered line, no arbitrary closer pair of points exists.
 
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