A collection of infinitely many distinct elements

[doronshadmi]

Let A be a rule

Let S be a non-finite collection of objects that obeys A.


An example:

S={a,b,c,d,e,f,g,…}

There is a bijection from S to itself (order is not important) for example:

a <--> b
b <--> g
c <--> a
d <--> d
e <--> c
f <--> f

…

Also we can show that there is no bijection between S to itself (order is not important) for example:

a <--> b
  <--> g
b <--> a
  <--> d
c <--> c
  <--> f

…

There are infinitely many maps from S to itself that are not a bijection, and a one case (again, order in not important) of a bijection between S to itself.

Why only this self mapping bijection case is considered by Standard Mathematics?

EDIT:

There is no single map of bijection. That is why a wrote that order is not important. But Standard map cares only about the bijection of S to itself and ignores the non-bijunction of S to itself.

I think that this non-bijanction is a property of a non-finite collection of distinct objects exactly as a proper subset of S is in a bejection with S (as a property of a non-finite collection of distinct objects) according to Standard Math.

Standard Math accepts the notion of a bijection between S and its proper subset, but ignores the non-bijanction mapping of S to itself as a proprty of a non-finite collection of distinct objects.

So my question is, why?

 

[fuelair]

Fortunately, as a not-mathematician, I am not compelled to be concerned. Some of our talented math members will be delighted to look over or overlook as appropriate! The question tends to make me suspicious that you believe you have an answer.

 

[Don At Work]

 

[nathan]

Let A be a rule

Let S be a non-finite collection of objects that obeys A.

What has A got to do with anything? AFAICT all we need here is a non-finite collection of objects.

Also we can show that there is no bijection between S ot itself (order is not important) for example:

What is 'ot'?

There are infinitely many maps from S to itself that are not a bijection, and a one case (again, order in not important) of a bijection between S to itself.

Why only this self mapping bijection case is considered by Standard Mathematics?

Why do you think there is only one bijection? Aren't there an infinite number of bijections?

There's only one identity bijection, where each element maps onto itself. Is that what you mean? But that's not the example bijective map you give.

 

[doronshadmi]

What has A got to do with anything? AFAICT all we need here is a non-finite collection of objects.

Maybe, but I mean to some class (A is some class).

What is 'ot'?

Typo. It has to be 'to'.

Why do you think there is only one bijection? Aren't there an infinite number of bijections?

There's only one identity bijection, where each element maps onto itself. Is that what you mean? But that's not the example bijective map you give.

There is no single map of bijection. That is why a wrote that order is not important. But Standard map cares only about the bijection of S to itself and ignores the non-bijunction of S to itself.

I think that this non-bijanction is a property of a non-finite collection of distinct objects exactly as a proper subset of S is in a bejection with S (as a property of a non-finite collection of distinct objects) according to Standard Math.


Standard Math accepts the notion of a bijection between S and its proper subset, but ignores the non-bijanction mapping of S to itself as a proprty of a non-finite collection of distinct objects.

So my question is, why?

I am gonig to add the last part of this post to my first post.

 

[Apathia]

Nice Doron.
I think I'll be able to muck about in this thread.
I'll get to it this evening after work.

 

[nathan]

Typo. It has to be 'to'.

Ok, your original post seems to be self contradictory then.

There is a bijection from S to itself (order is not important)
...

Also we can show that there is no bijection between S to itself (order is not important)

Please clarify this contradiction. Are you defining 'from' and 'between' in some way that is significant here? I cannot see what it might be.

 

[GreedyAlgorithm]

Hm, not true, if I understand you correctly. I assume by "order is not important" you mean "feel free to relabel the variables"? So that these two would be identical?
{a->b, b->c, c->a, d->d, e->e, f->f, ...} and
{a->c, b->a, c->b, d->d, e->e, f->f, ...}

But isn't this different than
{a->a, b->b, c->c, d->d, e->e, f->f, ...}

It seems to me I've shown an example of two different bijections. If not, explain why.

And as to non-bijections being "not considered by Standard Mathematics" or whatever, well, that's simply false. Or what you're thinking of is true but you've completely failed to communicate what you're thinking.

 

[doronshadmi]

It seems to me I've shown an example of two different bijections.

By "order is not important" I mean that it can be ordered or not.

As for your "not true" , it is clealry seen at post 1 that non-bijection of S to itself is a property of S in addition to a bijection of S to itself, and this non-bijection property of S to itself is the "sister" of a bijection between S to its proper subset.

 

[doronshadmi]

And as to non-bijections being "not considered by Standard Mathematics" or whatever, well, that's simply false.

No it is not false, as can be clearly seen at post one.

 

[doronshadmi]

Nice Doron.
I think I'll be able to muck about in this thread.
I'll get to it this evening after work.

Apathia,

As long as you get the organic paradigm as something that belongs to a particular framework like sinece or religion or what so ever, you simply miss it.

 

[nathan]

No it is not false, as can be clearly seen at post one.

Well, given that post one contradicts itself by saying there both is and is not a bijection, it's hard to say what post one shows.

 

[nathan]

As for your "not true" , it is clealry seen at post 1 that non-bijection of S to itself is a property of S in addition to a bijection of S to itself, and this non-bijection property of S to itself its the "sister" of a bijection between S to its proper subset.

This sentence is too tortured to be comprehensible. Can you break it apart to several smaller sentences?

 

[doronshadmi]

Well, given that post one contradicts itself by saying there both is and is not a bijection, it's hard to say what post one shows.

The contradiction is only in your mind.

S has both bijectionn and non-bijection with itsef, as clealy shown at post 1.

 

[doronshadmi]

This sentence is too tortured to be comprehensible. Can you break it apart to several smaller sentences?

Some particular example of a bijection between S and some proper subset of it:

a <--> a
b <--> c
c <--> e
d <--> g
e <--> i
f <--> k

…

Some particular example of a non-bijection between S and itself:

a <--> b
  <--> g
b <--> a
  <--> d
c <--> c
  <--> f

…

They are both properties of a non-finite collection of distinct objects.

 

[Apathia]

Apathia,

As long as you get the organic paradigm as something that belongs to a particular framework like sinece or religion or what so ever, you simply miss it.

Doron,

As a Liberal Arts Basketcase, I am even more interdisciplanary than you.
But I'm also a stickler about the rules and conventions of this forum.
I don't want to see your post moved or even dumped, because I drew the topic off to religion, as i so much want to do.

 

[Molinaro]

Bijection is a property that applies to or does not apply to a rule. It is not a property of a set. You seem to be saying it is a property of the set and that's simply wrong. The only way you can say the rule is a bijection on a particular set or not a bijection on a particular set is if you state the rule and the set in question.

Simply making a nonsense example and calling it both a bijection and non-bijection without stating the rule is completely meaningless (to me).

 

[Third Eye Open]

Let ╩ be a pirate.

Let ╫ Be the non-local ur-line.

Any n elements on the ur-line are represented by a, b, c ...

The length of the pirates sword can be found by this formula:

{{{╩}+{{╫}}}b/0(l)(L){o}|c|[k]/s\}= ◄ where ◄ is the length in microns.

If you don't understand this then you obviously haven't drilled a hole in your skull and sucked the non-local bits out with a straw. Once you do this you will be able to read this post with clarity.

 

[Soapy Sam]

What does "non-finite" mean?


In English, please.

 

[fuelair]

Doron,

As a Liberal Arts Basketcase, I am even more interdisciplanary than you.
But I'm also a stickler about the rules and conventions of this forum.
I don't want to see your post moved or even dumped, because I drew the topic off to religion, as i so much want to do.

OR, "Doron I wish you to the cornfield!!! You're not nice!!"