A collection of infinitely many distinct elements

[ddt]

Let A be a rule

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

In a later post you said A is a class, not a rule. Now I'm confused. What is it? And when you say class, do you mean class in the NBG axiomatization of set theory?

An example:

S={a,b,c,d,e,f,g,…}

So S is countable? Why not just write 1, 2, 3, 4, ...?

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

Functions are not ordered, so there is not order.

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

…

Okay.

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

That's a contradiction - there either is, or there is not a bijection of S to itself. "tertium non datur" - the law of the excluded middle. We're doing classical logic here, right? Or are you an intuitionist? Sorry, I forgot that until yesterday, you hadn't read a bit about existing logic theories from real mathematicians.

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.

No, there are infinitely many bijections of S to itself. You start with one bijection; let's call it f. Then when you swap the values f maps two elements, say, a and b, to, you have another bijection - that differs from f. As you can find infinitely many pairs of members a and b from S, you can do the swap trick in infinitely many ways. That all these bijections differ from each other is straightforward and left as an exercise to the reader.

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

No it isn't. Standard Math does study bijections because they are special - they have extra properties so you can derive more theorems about them than about ordinary functions.

As an example of a non-bijection that is considered by math, let me introduce you to a concept of your favourite mathematician Hilbert.

Hilbert had a special hotel (the Hilbert Hotel) with a (countably) infinite number of rooms - numbered 1, 2, 3, 4, ... Hilbert also operated special buses with infinitely many seats. One day, the Hilbert Hotel was full of guests, and then a full Hilbert bus arrived. Now how to accommodate those newly arrived guests?

Hilbert re-allocated all existing guests to other rooms. The guest in room #1 had to move to #2, the guest in room #2 to room #4, etc.: every guest moved to the room with the double number of his/her original room. So, after that, all odd-numbered rooms were free for the new guests from the bus.

The re-allocation function (the mapping from old room # to new room #) is clearly a non-bijective function.

 

[ddt]

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.

Sure. Still living in cuckoo-lala-land? In over 600 posts in the "Value in Logic\Mathematics" thread, you've not been able to describe what your "organic paradigm" is. You're going to do now?

BTW, what is "sinece"???

 

[nathan]

The contradiction is only in your mind.

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

No, the contradiction is with your use of language. 'there is a non-bijection' is a different claim to 'there is not a bijection'. The former claims the existence of something that is not a bijection. The latter claims the non-existence of anything that is a bijection.

It appears your original claim is that mappings exist, one of which is a bijection and the remainder of which are non-bijections.

That is false, as has already been pointed out. In a non-finite set, there are an infinite number of distinct bijections, even if one reorders symbols.

ETA:Oh, and that reordering of symbols would be what? Another bijection, perchance?

 

[nathan]

What does "non-finite" mean?


In English, please.

I consider it a synonym for (an unspecified) infinity. In this context Aleph₀.
Doron doesn't like infinities so I guess is using 'non-finite' as a circumlocution for the same concept. But perhaps Doron would like to define it?

 

[nathan]

You're contradicting yourself again.

There ... 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?

Claiming there is exactly one bijection.

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

Claiming there is at least one bijection.

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

Yeah, the set of possible mappings of a set onto itself can contain both bijective and non-bijective mappings. What of it?

 

[doronshadmi]

As a Liberal Arts Basketcase, I am even more interdisciplanary than you.

Good,

So please go beyond any particular discipline in order to be accurate about the organic paradigm.

 

[doronshadmi]

No, there are infinitely many bijections of S to itself.

And also infinitely many non-bijections of S to itself.

Here is some example:

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

…

Why do you ignore the non-bijections of S to itself?

 

[doronshadmi]

You're contradicting yourself again.



Claiming there is exactly one bijection.



Claiming there is at least one bijection.


Yeah, the set of possible mappings of a set onto itself can contain both bijective and non-bijective mappings. What of it?

Look Nathan,

If you wish to argue with youself, then go for it.

But I am not there.

 

[doronshadmi]

Sure. Still living in cuckoo-lala-land? In over 600 posts in the "Value in Logic\Mathematics" thread, you've not been able to describe what your "organic paradigm" is. You're going to do now?

BTW, what is "sinece"???

Your ignorence of the non-bijenctions of S to itself as a property of a non-finite collection of distinct objects (acctually you agree with its "sister", which is: a bijection of S with any given proper subset of it) is a good example of your inability to get new notions.

I can write ifnitiely many posts, but your ignorence pervents from you to get any one of them.

 

[nathan]

And also infinitely many non-bijections of S to itself.

Why do you ignore the non-bijections of S to itself?

I'm not. You claim there are an infinite number of non-bijections. I agree. We can move on for that point. You claim there is exactly one bijection in one post and at least one in another post. The former claim I disagree with. You need to clarify which claim your making. If you're now claiming the latter, that there is at least on bijection, I agree. We can move on. The question becomes, 'yes, what's your point?'

 

[nathan]

Look Nathan,

If you wish to argue with youself, then go for it.

Avoidance noted.

But I am not there.

I think you're demonstrating that remarkably well

 

[nathan]

Your ignorence of the non-bijenctions of S to itself as a property of a non-finite collection of distinct objects (acctually you agree with its "sister", which is: a bijection of S with any given proper subset of it) is a good example of your inability to get new notions.

As I pointed out, this sentence is too tortured to understand. Your explanation of it was to claim there are both bijective and non-bijective mappings. What of it? What's your point? You claim 'Standard Maths' ignores the non-bijective mappings, but that's completely false. As has been pointed out, Hilbert's Hotel relies on a non-bijective mapping. Any function with both positive and negative gradients is a non-bijective mapping of the real numbers. Standard Maths studies a lot of those.

If you fail to make some kind of coherent claim or point in your very next post I vote moving to AAH

 

[zosima]

@Doron

Bijections have many special properties. So they are often more interesting and easier to deal with. That would explain why they are often featured in many theorems, particularly in introductory texts. Bijections are isomorphisms, this means that structure is preserved across transformations. Systems with this sort of regularity and internal structure often allow a mathematician to "get further" in terms of the results they produce.

I don't think you should take this as evidence that mathematicians aren't interested in non-bijections. I think you should trust me when I say that both sorts of mappings are so basic that they are both used and investigated heavily. It seems to me that you perceive some sort of bias or prejudice in mathematics, what reason do you have to believe this? Do you have a reason? Or does this perception just follow from your anthropomorphized conception of the field of mathematics?

 

[doronshadmi]

I'm not. You claim there are an infinite number of non-bijections. I agree. We can move on for that point. You claim there is exactly one bijection in one post and at least one in another post.

No. There is not exactly one bijection.

There are infinitely many bijections, where each bijection has another order, but it does not matter because order is not important. GET IT???

We can move on for that point. If you continue to claim this, you speack to yourself.

 

[doronshadmi]

As I pointed out, this sentence is too tortured to understand. Your explanation of it was to claim there are both bijective and non-bijective mappings. What of it? What's your point? You claim 'Standard Maths' ignores the non-bijective mappings, but that's completely false. As has been pointed out, Hilbert's Hotel relies on a non-bijective mapping. Any function with both positive and negative gradients is a non-bijective mapping of the real numbers. Standard Maths studies a lot of those.

If you fail to make some kind of coherent claim or point in your very next post I vote moving to AAH

Hilbert's Hotel is not relevant here because I am talking about a mapping of S to itself. No new objects (as can be found at Hilbert's Hotel analogy) are involved at the mapping of S to itself. GET IT ??

 

[doronshadmi]

@Doron

Bijections have many special properties. So they are often more interesting and easier to deal with. That would explain why they are often featured in many theorems, particularly in introductory texts. Bijections are isomorphisms, this means that structure is preserved across transformations. Systems with this sort of regularity and internal structure often allow a mathematician to "get further" in terms of the results they produce.

I don't think you should take this as evidence that mathematicians aren't interested in non-bijections. I think you should trust me when I say that both sorts of mappings are so basic that they are both used and investigated heavily. It seems to me that you perceive some sort of bias or prejudice in mathematics, what reason do you have to believe this? Do you have a reason? Or does this perception just follow from your anthropomorphized conception of the field of mathematics?

Please direct us to some mathematical work where S has no bijection with itself.

 

[nathan]

No. There is not exactly one bijection.

ok, glad you agree.

There are infinitely many bijections, where each bijection has another order, but it does not matter because order is not important. GET IT???

What do you mean? Do you claim all these bijections are the same in some way? What way? As near as I can make out, because all these bijections map the set X to the same set X, you claim they're 'the same'. Is that what you claim?

 

[nathan]

Hilbert's Hotel is not relevant here because I am talking about a mapping of S to itself. No new objects (as can be found at Hilbert's Hotel analogy) are involved at the mapping of S to itself. GET IT ??

You must have a different definition of non-finite then. Hilbert's hotel does involve a non-bijective mapping of the postive integers (a non-finite set) onto itself. Where do you think the new objects are? No new hotel rooms are created (these are the members of the set Hilbert is discussing). That's Hilbert's whole point -- the hotel is full, but can always accommodate new guests (even an infinite number of new guests). Unfortunately bell-hops are always moving your luggage around

Oh, and shouting about things doesn't make them any truer.

 

[nathan]

Any function with both positive and negative gradients is a non-bijective mapping of the real numbers. Standard Maths studies a lot of those.

I wish to point out this is not true. any continuous function ...
I was attempting to simplify a description of bijections of R, and flubbed it. Anyway, my point still stands that standard maths is quite happy with non bijective functions.

 

[nathan]

double