Deeper than primes

[epix]

You are taking the wrong way.

Please read very carefully http://www.internationalskeptics.com/forums/showpost.php?p=7083010&postcount=15043 including its links, in order to understand my arguments.

I think I got it.

 

[doronshadmi]

I think I got it.

[qimg]http://www.pattayanewbie.com/Images/thai%20electrical%20plugs.jpg[/qimg]


[qimg]http://comps.fotosearch.com/comp/AGE/AGE021/two-pigs-snouts_~K30-331525.jpg[/qimg]

I like your creative imagination.

 

[punshhh]

No. It is absolutely the smallest existing element. Actually is the totally finite existing element.

So the point has no length like a line segment which has the property of length.

As a totally accurate location.

Do you rule out an infinitely small point in that precise location?
If so why?

No smallest element exists at once in more than one location, which is a property that an ever smaller element has.

Ok, can there be more than one smallest element?
Can there be less than one smallest element (if nothing exists)?

 

[epix]

The standard proof for Cantor's Theorem posits a bijection between the elements of S and P(S). Yours is not a bijection, just a random injection from S to P(S).

It does appear to be random. He prompted me again, so I followed his choices that include the empty sets.

|-----------------------
|a ↔ { } |
|b ↔ {b} | Provides {a}
|c ↔ {c} |
|-----------------------

a ↔ {a} |
|b ↔ { } | Provides {b}
|c ↔ {c} |

|-----------------------
|a ↔ {a} |
|b ↔ {b} | Provides {c}
|c ↔ { } |
|-----------------------

|-----------------------
|a ↔ { } |
|b ↔ {c} | Provides {a,b}
|c ↔ {a,c} |
|-----------------------

|-----------------------
|a ↔ { } |
|b ↔ {b} | Provides {a,c}
|c ↔ {a,b} |
|-----------------------

|-----------------------
|a ↔ { } |
|b ↔ {a} | Provides {a,b,c}
|c ↔ {b} |
|-----------------------

He links { } with a, b, c, which provides {a}, {b}, {c}, and then he links it with a, a, a, where he breaks the pattern in the last configuration. I hope he's okay otherwise.

 

[jsfisher]

So according to jsfisher if S = {A,B} then P(S) = {{},{A,B}}.

No, that is your reasoning.

...The right reasoning is this:

S = {A,B}

P(S) = {{},{A},{B},{A.B}}

Cantor's constriction method of all the above P(S) members

Umm, you already have all of the P(S) members. There's nothing more that needs to be done.

...
|-----------------------
|a ↔ {a} |
|b ↔ {b} | Provides {}
|-----------------------

Why this injection? You followed no stated rule. You just picked this mapping. Why? What makes your arbitrary choices for the 4 injections special?

 

[epix]

Let us demonstrate Cantor's construction method, by using |P(S)| proper subsets of P(S) members, where each P(S) proper subset has |S| different P(S) members that are mapped with all S members.

In this demonstration we are using the all members of S={a,b,c} and |P(S)| proper subsets of different members of P(S)={{},{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}}, where each used P(S) proper subset has |S| different P(S) members, as follows:

.
.
.
|
|
|-----------------------
|a ↔ {a} |
|b ↔ {b} | Provides {}
|c ↔ {c} |
|-----------------------
|
|
|-----------------------
|a ↔ { } |
|b ↔ {b} | Provides {a}
|c ↔ {c} |
|-----------------------
|
|
|-----------------------
|a ↔ {a} |
|b ↔ { } | Provides {b}
|c ↔ {c} |
|-----------------------
|
|
|-----------------------
|a ↔ {a} |
|b ↔ {b} | Provides {c}
|c ↔ { } |
|-----------------------
|
|
|-----------------------
|a ↔ { } |
|b ↔ {c} | Provides {a,b}
|c ↔ {a,c} |
|-----------------------
|
|
|-----------------------
|a ↔ { } |
|b ↔ {b} | Provides {a,c}
|c ↔ {a,b} |
|-----------------------
|
|
|-----------------------
|a ↔ {a} |
|b ↔ {c} | Provides {b,c}
|c ↔ {a,b} |
|-----------------------
|
|
|-----------------------
|a ↔ { } |
|b ↔ {a} | Provides {a,b,c}
|c ↔ {b} |
|-----------------------
|
|
.
.
.

As can be seen, we have constructed the all P(S) members without exceptional, by using Cantor's construction method, and it does not matter if |P(S)| and |S| are finite or infinite.

In the case where |P(S)| and |S| are infinite, Dedekind's infinite holds (where S is a proper subset of P(S)).

I don't understand any of it. Is "↔" a symbol for a function? It's usage above implies that it isn't, coz you cannot have one member of set A pointing to more than one member of set B, which is clearly happening above, unless the triples above mean

A -> B
A -> C
A -> D
A -> E

and so on. If this possibility is true, what are the sets B, C, D, E, ... ?

 

[epix]

You are taking the wrong way.

No, I don't. After completing this module



you will be able to tell apart injection from surjection in this exercise:

 

[dafydd]

As I see it an 0 dimensional point has no size or diameter, it is infinitely small. If in our attempt to define a point(A) on a line segment, we have an infinite regress into an ever smaller point. accompanied by an infinite regression of ever smaller line segments. We can have no defined length at any point on this line segment. If we do define a length(N), this length would be infinitely longer not only in relation the size of the point on it, but also against the length of an ever shorter line segment(S), as we proceed on our infinite regress.

Now if in our infinite regress towards the point, we were to turn around and attempt to define the length of N, we would then discover that it is infinitely longer than the line segment we were at when we turned around.

Hence you can never have a value for N because;

whatever value you give to N, it is infinitely longer than A and S
Also whatever value you give to S, it is infinitely shorter than N.

Between S and N is an infinite sliding scale of value.

So if you attempt to give a value to S(S1) it is infinitely larger than S and is equivalent to N.
If you attempt to give a value to N(N1) it is also infinitely larger than S and is equivalent to S1.

S and N are both potentially infinitely longer than S1 and N1

If N is infinitely long A could also be infinitely long and still remain infinitely short.

Point A is both infinitely short/small and infinitely long/large.

No. Please don't bother your head with things that you don't understand.

 

[dafydd]

doronshami and punshhh in the same thread,it's a dream come true. Talk about the blind leading the blind.

 

[epix]

As I see it an 0 dimensional point has no size or diameter, it is infinitely small.

I think that if something has no size, then it actually doesn't exist in the context discussed. (A thought doesn't have a "size," but it does exist.) I think point is like your index, which you point toward things of your interest, but that finger is never a part of the objects you're pointing toward, even if your finger comes into contact with that object. When you select an object, you use your index to point toward certain locations on it. I think that point answers the question where? with there.

Things are usually defined with respect to where they appear, and so here is one of the definitions of spatial point:

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points have neither volume, area, length, nor any other higher dimensional analogue. Thus, a point is a 0-dimensional object. In branches of mathematics dealing with set theory, an element is often referred to as a point. A point could also be defined as a sphere which has a diameter of zero.

Note the term "primitive notion" and its explanation:

In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience.

That means you are sort of free to define points for yourself, if that concept seems to be present in some special environment that you investigate.

 

[doronshadmi]

doronshami and punshhh in the same thread,it's a dream come true. Talk about the blind leading the blind.[qimg]http://www.internationalskeptics.com/forums/imagehosting/234094da7c17504403.gif[/qimg]

dafydd, What You "See" Is What You "Get" (W.Y."S".I.W.Y."G" in your case (http://en.wikipedia.org/wiki/WYSIWYG))

For example, you are blind to posts like http://www.internationalskeptics.com/forums/showpost.php?p=7087283&postcount=15074 .

 

[doronshadmi]

No, that is your reasoning.



Umm, you already have all of the P(S) members. There's nothing more that needs to be done.



Why this injection? You followed no stated rule. You just picked this mapping. Why? What makes your arbitrary choices for the 4 injections special?

jsfisher, your last post demonstrates again that your used reasoning is too weak in order to comprehend the following posts:

http://www.internationalskeptics.com/forums/showpost.php?p=7079697&postcount=15030

http://www.internationalskeptics.com/forums/showpost.php?p=7087199&postcount=15073


As the people of your community like to say: "no less, no more". You exactly do not get it.

 

[doronshadmi]

I don't understand any of it. Is "↔" a symbol for a function? It's usage above implies that it isn't, coz you cannot have one member of set A pointing to more than one member of set B

The first stage of using "↔" is done by using |P(S)| mappings between S members and |S| amount of P(S) members, which produces the all P(S) members, without exceptional.

By using the fact that all P(S) members are constructed without exceptional, and by using the fact that both S and P(S) are infinite sets, we are able to show that Dedekind's Infinite holds between S and P(S), exactly as it holds between Q and N or between N and E (where E is the set of all even numbers), etc.

In other words, Dedekind's Infinite (http://en.wikipedia.org/wiki/Dedekind-infinite_set) is true for all infinite sets, without exceptional, and the notion of uncountable infinite sets is false.

--------------

You also still have to grasp http://www.internationalskeptics.com/forums/showpost.php?p=7083010&postcount=15043 .

 

[doronshadmi]

So the point has no length like a line segment which has the property of length.

Please explain it in details.

Do you rule out an infinitely small point in that precise location?
If so why?

A point is not infinitely small. A point is the smallest and totally finite element.

Ok, can there be more than one smallest element?

There can be infinitely many smallest and totally finite elements.

But no collection of such elements completely covers an element like a line segment.

Can there be less than one smallest element (if nothing exists)?

Nothing is less than a one smallest element, and Nothing is the total absence if Existence.

 

[doronshadmi]

Umm, you already have all of the P(S) members.

It is done by using a construction method, without missing even a single member of P(S), whether |P(S)| is finite or not.

There's nothing more that needs to be done.

There is.

We are able to show a bijection between all the members of the form a,b,c,d,e,... and all the members of the form {x}, where x is a placehoder for Nothing or any proper subset of set Z={a,b,c,d,e,...}.

 

[punshhh]

Please explain it in details.

A point is a position on a line which covers no part of the length of the line?
It only denotes a position on the line.

A point is not infinitely small. A point is the smallest and totally finite element.

This totally finite element is not present in the sense;

It has no length
no width
no height, ok?

It is bigger than an infinitely small point, but smaller than any other possible thing?

There can be infinitely many smallest and totally finite elements.

Yes can an infinite number of points occupy the same precise position?

But no collection of such elements completely covers an element like a line segment.

Yes, it would require an infinitely large number of ever smaller line segments to cover a line segment, which is impossible?

Nothing is less than a one smallest element, and Nothing is the total absence if Existence.

This suggests that your point is of the order of scale of one unit on the Planck scale

So there is no infinitely small element in existence?

Is there no infinitely large element?

Nothing is impossible, right?

 

[dafydd]

A point is a position on a line which covers no part of the length of the line?
It only denotes a position on the line.





This totally finite element is not present in the sense;

It has no length
no width
no height, ok?

It is bigger than an infinitely small point, but smaller than any other possible thing?




Yes can an infinite number of points occupy the same precise position?



Yes, it would require an infinitely large number of ever smaller line segments to cover a line segment, which is impossible?




This suggests that your point is of the order of scale of one unit on the Planck scale

So there is no infinitely small element in existence?

Is there no infinitely large element?

Nothing is impossible, right?

I cannot believe that you are taking doronshami seriously.

 

[laca]

I cannot believe that you are taking doronshami seriously.

Why not? It actually makes sense.

 

[dafydd]

Why not? It actually makes sense.

There's a first time for everything.

 

[jsfisher]

jsfisher, your last post demonstrates again that your used reasoning is too weak in order to comprehend the following posts....

You continue to try to divert attention from the earlier bogus claims you have made. You claimed |S| = |P(S)| for any set S, finite or infinite. You also claimed there to be a bijection between the elements of any set S and its power set.

Either retract these bogus claims, or perform the impossible by providing a bijection between the elements of the null set and its power set.

There will be plenty of time afterwards to deal with your other bogus claims about construction methods and all that. First things first, though.