Little 10 Toes
Master Poster
Then either you do not understand what |N| means, what || means, or what the set N is.
Besides, you used it in your example, so why can't I?
Deeper than primes - Continuation
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Then either you do not understand what |N| means, what || means, or what the set N is.
Besides, you used it in your example, so why can't I?
Kid Eager
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Nope, the logic in that thread does not follow. You *have* redefined the word, and by doing so, the integrity of the logic fails.
You admit as much in your "explanation".
doronshadmi
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Since Traditional Mathematics does not use a function without an inverse, it is tuned to get only |N|=|N|.
For better understanding of visual_spatial AND verbal_symbolic reasoning, let us use a common ground for our abstract mathematical reasoning, as follows:
The common ground is an unbounded grid (the terms domain and codiman are not used in that framework) where horizontal observation is used for distinction and vertical observation is used for comparison (distinction and comparison are possible in the first place exactly because they are non-local (they are at AND beyond a given locality)), as follows:
An example of a set with unbounded amount of members, that is compared with itself on the common ground:
Code:
The common ground |N|=|N| OR |N| ¬= |N|
... ... ...
| | | | | | | | | | | | | | | | | | | | |
_____________________ _____________________ _____________________
| | | | | | | | | | | | | | | | | | | | |
_____________________ ____1__2__3__4__5__6_ ____1__2__3__4__5__6_
... | | | | | | | ... ... | ↕ ↕ ↕ ↕ ↕ ↕ ... ... | ↓ ↕ ↕ ↕ ↕ ↕ ...
_____________________ ____1__2__3__4__5__6_ _______1__2__3__4__5_
| | | | | | | | | | | | | | | | | | | | |
_____________________ _____________________ _____________________
| | | | | | | | | | | | | | | | | | | | |
... ... ...An example of a proper subset with unbounded amount of members that is compared with its set on the common ground:
Code:
The common ground |P|=|N| OR |P| > |N|
... ... ...
| | | | | | | | | | | | | | | | | | | | |
_____________________ _____________________ _____________________
| | | | | | | | | | | | | | | | | | | | |
_____________________ ____2__4__6__8_10_12_ ____2__4__6__8_10_12_
... | | | | | | | ... ... | ↕ ↕ ↕ ↕ ↕ ↕ ... ... | ↓ ↕ ↕ ↕ ↕ ↕ ...
_____________________ ____1__2__3__4__5__6_ _______1__2__3__4__5_
| | | | | | | | | | | | | | | | | | | | |
_____________________ _____________________ _____________________
| | | | | | | | | | | | | | | | | | | | |
... ... ...By using the common ground, ↕ is a function with an inverse on the common ground, where ↓ (or ↑) is a function without an inverse on the common ground.
By using the common ground we immediately realize that Traditional Mathematics uses only functions with an inverse on the common ground, and us a result it can't handle with the other possible comparisons on the common ground (as done by Organic Mathematics, which uses the non-locality of the common ground (which enables the use of functions without inverse) for finer understanding of the localities and their possible relations on it).
In other words, the notion of domain and codomain is no more than the particular case of functions with an inverse on the common ground, which prevent finer understanding of the localities and their possible relations on the common ground.
The common ground is the non-local cross-context foundation of Organic Mathematics, which enables finer and richer understanding of the context-dependent localities and their possible relations on it.
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doronshadmi
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Please think about fog.
The not well-defined size of set with unbounded amount of members, is straightforward exactly as the identity of the fog is being unclear.
Kid Eager
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What the?
realpaladin
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I am reading that with the voice of Stewie... and it is quite appropriate in this context 
Little 10 Toes
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I'm getting tired of your assertions without any proof. Please provide direct proof that "traditional Mathematics does not use a function without an inverse".
<snip garbage>
I always love how you say something about a set isn't equal to itself, start doing some sort of table or graphic and yet never start the set at the same starting point.
I've pointed it out to you before. Yet you continue to make the same mistake.
realpaladin
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Doron will fail, because y=x^2 is a function without an inverse.
Or, if you want a more elaborate explanation: http://www.sosmath.com/algebra/invfunc/fnc1.html
doronshadmi
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EDIT:
The non-locality of ____ w.r.t . (___ is at AND beyond the one and only one position of . along it) or the locality of . w.r.t ____ (. is at one and only one given position along ____) is an axiom, which means it does not need any proof.
As long as you get things only in terms of locality you are unable to understand why a collection with unbounded amount of . does not have an accurate cardinality.
As long as you are using only verbal_symbolic reasoning that is focused only on the symbols of the numbers (which are localities along the non-local common ground, as shown in http://www.internationalskeptics.com/forums/showpost.php?p=9823863&postcount=3563) you simply have no way to understand why |N| = OR not= |N|.
In order to get it at list visual_spatial AND verbal_symbolic reasoning has to be used, where by this reasoning there is a mapping (a function) without an inverse on the non-local common ground among localities w.r.t their selves or w.r.t other localities, such that the non-local common ground does not domain and range (codomain) among functions.
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doronshadmi
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Please demonstrate how this function is used as a mapping between members of sets.
Please be aware that I am using a function as a mapping between members of sets, where instead of domain and codomain a non-local common ground is used among the member, as explained in http://www.internationalskeptics.com/forums/showpost.php?p=9823863&postcount=3563 and http://www.internationalskeptics.com/forums/showpost.php?p=9824985&postcount=3569.
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Little 10 Toes
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Your last post does not match anything to what I said. You have decided to invent yet another term that you don't need to use. When you are comparing two sets why aren't you starting with the first element of the sets?
Little 10 Toes
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Doron, please stop going back and editing your posts.
And there you have it, ladies and gents. Doron has no clue what a function is, yet he claims to have improved on all things maths! Ridiculous.
doron, the only person your inane gibberish appeals to is... wait for it... YOU! Doesn't that tell you something? With all the skills you claim to have that nobody else does, you can't see that you're spouting useless, retarded gibberish?
doronshadmi
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EDIT:
This is not the case with verbal_symbolic AND visual_spatial reasoning which enables to deal with the relations among locality and non-locality, and I did not invent it in this page so this is not another term, but it is an improved explanation of the relations among locality and non-locality, which can be understood only by posters that think in terms of abstract verbal_symbolic AND visual_spatial reasoning.
As you can see, I can start from any place (the first,not the first,last) and get the same result, exactly because the size of sets with bounded amount of members is well-defined.
The same approach (the first, not the first) is used among sets with unbounded amount of members, but since they have no last members, the result is |N| = OR not= |N|, or in other words, unlike in the case of sets with bounded amount of members, in the case of sets with unbounded amount of member (whether it is one and only one set, or not) I do not get the same results by using the first or not the first. In other words the size of sets with unbounded amount of members is not well-defined, for example:
OR
You have asked about proofs, but axioms do not need proofs.
It is not used by verbal_symbolic-only reasoning that understands thing only in terms of locality.
This is not the case with verbal_symbolic AND visual_spatial reasoning which enables to deal with the relations among locality and non-locality, and I did not invent it in this page so this is not another term, but it is an improved explanation of the relations among locality and non-locality, which can be understood only by posters that think in terms of abstract verbal_symbolic AND visual_spatial reasoning.
Take, for example, the following mapping among sets with bounded amount of members:
Code:
{1,2,3,4,5}
↓ ↕ ↕ ↕ ↕
{1,2,3,4}As you can see, I can start from any place (the first,not the first,last) and get the same result, exactly because the size of sets with bounded amount of members is well-defined.
The same approach (the first, not the first) is used among sets with unbounded amount of members, but since they have no last members, the result is |N| = OR not= |N|, or in other words, unlike in the case of sets with bounded amount of members, in the case of sets with unbounded amount of member (whether it is one and only one set, or not) I do not get the same results by using the first or not the first. In other words the size of sets with unbounded amount of members is not well-defined, for example:
Code:
{1,2,3,4,5,...}
↓ ↕ ↕ ↕ ↕
{1,2,3,4,...}OR
Code:
{1,2,3,4...}
↕ ↕ ↕ ↕
{1,2,3,4,...}
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doronshadmi
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You are right, I added "EDIT:".
Wow, you got one right! OK, axioms do not need proof, indeed. They do need to be useful, however, not to mention consistent. Care to show some useful result arrived at by starting from your axioms? Didn't think so... We all know you got exactly nothing, doron, and you keep on reasserting the same.
realpaladin
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That is just a plain silly statement; the function speaks for itself. It is the most simple function possible. Even grade-school kids understand this one.
Besides having proven you wrong, I have also proven you a liar, Doron Shadmi.
It was you who claimed traditional mathematics had no functions without an inverse.
And now you need to drag everything-and-the-kitchensink in to make it stick. And even then you fail.
Failure hath a name; it is Doron Shadmi.
realpaladin
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I will add the whole Island A/Island B debacle to Doron Shadmi's list of errors, as well as his hilarious attempt to backpedal on "Traditional mathematics has no functions without an inverse" and of course the classic "Show that y=x^2 can be used as a mapping between two sets".
The new 'common ground' blahblah that Doron tries to introduce is something the rest of the world already knows as a grid (for ages). This is also why I used the x and y in the example; if he really had spatial_visual reasoning he would immediately have recognized that function.
But he hasn't, so he didn't
doronshadmi
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In addition to my previous post (http://www.internationalskeptics.com/forums/showpost.php?p=9825141&postcount=3574), please look at this comparison, that is done among one and only one set, which is the set of all natural numbers.
Code:
|{1,2,3,4,...}|
↑ ↑ ↑ ↑ ↕ ↕ ↕ ↕
|{...,7,5,3,1,2,4,6,8,...}|As you can see the set of all natural numbers does not necessarily have a first member (in addition to not having last member), so in this case there are members that have functions without an inverse, which prevent the bijection from N into itself.
Here is an example of a set with unbounded amount of members, that is compared with itself on the common ground (no domain and codomain are used here):
Code:
The common ground |N| ¬= |N|
... ...
| | | | | | | | | | | | | |
_____________________ _____________________
| | | | | | | | | | | | | |
_____________________ __________1__2__3__4_
... | | | | | | | ... ... ↑ ↑ ↑ ↕ ↕ ↕ ↕ ...
_____________________ _5__3__1__2__4__6__8_
| | | | | | | | | | | | | |
_____________________ _____________________
| | | | | | | | | | | | | |
... ...By using the common ground (which its horizontal aspect is distinction, and its vertical aspect is comparison among distinct objects, where both distinction and comparison are non-local w.r.t the given objects) we do not use domain and codomain, because the common ground is the one and only one domain for all comparisons among all the distinct objects.
This common ground is not the Cartesian X/Y grid, as used, for example, by realpaladin in its y=x^2 expression, but it is the common ground of the relations among locality and non-locality (which is not restricted to Geometry or Metric space).
More details about the common ground are given in http://www.internationalskeptics.com/forums/showpost.php?p=9823863&postcount=3563.
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doronshadmi
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This is exactly the case among functions that are used to compare given sets in order to find if their sizes are the same or not the same.
Again, please show how y=x^2 (which has no inverse function) is used in order to determine if two given sets have or do not have the same size (where also the case of set's self comparison is included).
I am waiting for you detailed demonstration.
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