A collection of infinitely many distinct elements

[nathan]

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

What is your point? No one's claiming there are no bijections of a set onto itself. That's a ridiculous claim. The identity mapping is a bijection. It can apply to any set. Ergo every set has at least one bijective mapping to itself.

 

[doronshadmi]

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.

Ok.

At a mapping of S to itself there are no new rooms and no new guests, so Hilbert's analogy is not a mapping of S to itself.

 

[doronshadmi]

The identity mapping is a bijection.

Good.

But I clearly show that there are also mappings that are non-bijection from S to itself.

Again, here is some example:

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

…

Why do you ignore mappings that are non-bijection from S to itself?

What is your point?

 

[ddt]

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

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

I gave one example of a mapping of an infinite set, viz. N, to itself which is not a bijection (it is a injection, a.k.a. 1-to-1, of course) in the Hilbert Hotel story.

Actually, there were already 2 non-bijective functions in that case:

1. the re-allocation of room numbers for the existing guests
2. the mapping of the new guests to their rooms: for the new guests, on entering the bus on departure of their trip, each got a number allotted by the tour operator

The story of Hilbert's Hotel is not yet over. Hilbert was indeed a very special tour operator: not only did each of his buses have infinitely many seats, he also had infinitely many buses.

As an aside: this of course makes price fighters such as Richard Branson green with envy. Think of the profits you can make with such buses! Branson has tried to recreate Hilbert's success as tour operator. Needless to say, he failed. The phenomenon has only been observed thus far in the beginning of the 20th C. in central Germany, a bit south-east of Hanover. The only explanation thus far is that the extremely high density of mathematicians and physicists created a rift in the space/time continuum which made this possible. The phenomenon disappeared instantly after the Nazi's purged many of those because they were jewish.

So one day, infinitely many full Hilbert buses (so with each infinitely many passengers) arrive at Hilbert's Hotel, which is also full at that moment with infinitely many guests. How to accommodate all those newly arrived guests?

For the hotel manager Hilbert, however, this is no problem. He already has one non-bijective function p which enumerates all prime numbers; so p(1) = 2, p(2) = 3, p(3) = 5, etc. As a true manager, Hilbert delegates the actual calculation of this function to an assistant: Eratosthenes, the cook, who in the kitchen is busy with infinitely many sieves to sieve the prime numbers from all natural numbers.

Now Hilbert moves the already established guests according to function p; so the guest in room #1 moves to room #2, the guest in room #2 moves to #3, etc. This makes room for the newly arrived guests. Now, Eratosthenes' sieving has as interesting by-product the creation of infinitely many infinite sequences.

Eratosthenes' first sieve has intercepted all multiples of 2; so we can make a function f_1 which lists all these numbers in ascending order (so f_1(1) = 4, f_1(2) = 6, f_1(3) = 8, etc). The guests who arrive in bus #1 (of course, the buses are also numbered) are allocated to rooms according to this function.

Eratosthenes' second sieve has intercepted all multiples of 3 that are not a multiple of 2 (the multiples of 2 are stuck in the first sieve, remember); so we can make a function f_2 which lists all these numbers in ascending order (so f_2(1) = 9, f_2(2) = 15, f_2(3)=21, etc.). The guests who arrive in bus #2 are allocated to rooms according to this function.

We can continue this process for any number. Eratosthenes' n-th sieve intercepts all multiples of the n-th prime which are not multiple of a lower-numbered prime. This constructs a function f_n which Hilbert uses for allocating the guests from bus #n. So passenger #m from bus #n gets room number f_n(m).

As you can see, all functions f_n are injective (1-to-1) but not surjective (onto), so they're all not bijective. We have created infinitely many non-bijective functions from the infinite set of natural numbers to itself.

The careful reader may also have noted that this leaves one room free: room #1. Together with the arrival of the Hilbert buses, a car came with a single guest from Laren, Netherlands. This guest, Brouwer, in his younger years had seen and supported the building of Hilbert's empire, but had come to doubt it could exist in the first place, so he came back to discuss with Hilbert about that. Of course, Hilbert gave him the first and biggest room in his hotel, where they would argue about the existence of the hotel.

Later, another anecdote would surface. Hilbert had a young English receptionist, fresh from hotel school, named Cleese. He didn't get along with Hilbert and some of his assistants - the German ones. The day after Brouwer arrived, he also put on the prank to stick an extra "2" on the door of Brouwer's hotel room.

 

[ddt]

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.

No, I don't get it either. There is no "order". Bijections deal with arbitrary sets, without looking at structure - so they don't consider an ordering structure either.

There is no inherent order in the set of bijections either.

 

[ddt]

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.

A small correction: an isomorphism is a bijection, but not every bijection is an isomorphism. Bijections only deal with "bare" sets without any structure; isomorphisms deal with sets with some structure on it. So a group isomorphism is a bijection that preserves the group structure (addition and neutral element), a field isomorphism preserves the field structure, an ordering isomorphism preserves the ordering structure, etc.

 

[doronshadmi]

I gave one example of a mapping of an infinite set, viz. N, to itself which is not a bijection (it is a injection, a.k.a. 1-to-1, of course) in the Hilbert Hotel story.

Actually, there were already 2 non-bijective functions in that case:

1. the re-allocation of room numbers for the existing guests
2. the mapping of the new guests to their rooms: for the new guests, on entering the bus on departure of their trip, each got a number allotted by the tour operator

The story of Hilbert's Hotel is not yet over. Hilbert was indeed a very special tour operator: not only did each of his buses have infinitely many seats, he also had infinitely many buses.

As an aside: this of course makes price fighters such as Richard Branson green with envy. Think of the profits you can make with such buses! Branson has tried to recreate Hilbert's success as tour operator. Needless to say, he failed. The phenomenon has only been observed thus far in the beginning of the 20th C. in central Germany, a bit south-east of Hanover. The only explanation thus far is that the extremely high density of mathematicians and physicists created a rift in the space/time continuum which made this possible. The phenomenon disappeared instantly after the Nazi's purged many of those because they were jewish.

So one day, infinitely many full Hilbert buses (so with each infinitely many passengers) arrive at Hilbert's Hotel, which is also full at that moment with infinitely many guests. How to accommodate all those newly arrived guests?

For the hotel manager Hilbert, however, this is no problem. He already has one non-bijective function p which enumerates all prime numbers; so p(1) = 2, p(2) = 3, p(3) = 5, etc. As a true manager, Hilbert delegates the actual calculation of this function to an assistant: Eratosthenes, the cook, who in the kitchen is busy with infinitely many sieves to sieve the prime numbers from all natural numbers.

Now Hilbert moves the already established guests according to function p; so the guest in room #1 moves to room #2, the guest in room #2 moves to #3, etc. This makes room for the newly arrived guests. Now, Eratosthenes' sieving has as interesting by-product the creation of infinitely many infinite sequences.

Eratosthenes' first sieve has intercepted all multiples of 2; so we can make a function f_1 which lists all these numbers in ascending order (so f_1(1) = 4, f_1(2) = 6, f_1(3) = 8, etc). The guests who arrive in bus #1 (of course, the buses are also numbered) are allocated to rooms according to this function.

Eratosthenes' second sieve has intercepted all multiples of 3 that are not a multiple of 2 (the multiples of 2 are stuck in the first sieve, remember); so we can make a function f_2 which lists all these numbers in ascending order (so f_2(1) = 9, f_2(2) = 15, f_2(3)=21, etc.). The guests who arrive in bus #2 are allocated to rooms according to this function.

We can continue this process for any number. Eratosthenes' n-th sieve intercepts all multiples of the n-th prime which are not multiple of a lower-numbered prime. This constructs a function f_n which Hilbert uses for allocating the guests from bus #n. So passenger #m from bus #n gets room number f_n(m).

As you can see, all functions f_n are injective (1-to-1) but not surjective (onto), so they're all not bijective. We have created infinitely many non-bijective functions from the infinite set of natural numbers to itself.

The careful reader may also have noted that this leaves one room free: room #1. Together with the arrival of the Hilbert buses, a car came with a single guest from Laren, Netherlands. This guest, Brouwer, in his younger years had seen and supported the building of Hilbert's empire, but had come to doubt it could exist in the first place, so he came back to discuss with Hilbert about that. Of course, Hilbert gave him the first and biggest room in his hotel, where they would argue about the existence of the hotel.

Later, another anecdote would surface. Hilbert had a young English receptionist, fresh from hotel school, named Cleese. He didn't get along with Hilbert and some of his assistants - the German ones. The day after Brouwer arrived, he also put on the prank to stick an extra "2" on the door of Brouwer's hotel room.

You don't get it do you?

In a non-bijection of S to itself there are no guests who arrive because there are no steps called buses here.

 

[ddt]

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 ...

It actually is true. When you speak of gradients, you imply the function is differentiable, and any differentiable function is continuous.

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

I second that. Doron would be well-advised to pick up some good introductory texts on basic math and find a good tutor to coach him/her with it.

 

[ddt]

You don't get it do you?

In a non-bijenction of S to itself there are no guests who arrive because there are no steps called buses here.

The guests are immaterial to the point. The point is that I construct infinitely many non-bijections from N to itself.

 

[nathan]

Why do you ignore mappings that are non-bijection from S to itself?

I'm ignoring them in this discussion because we don't disagree about them. Go back and read post 30.

What is your point?

I am correcting the mistakes in your claims, and attempting to clarify what you are claiming.

 

[doronshadmi]

No, I don't get it either. There is no "order". Bijections deal with arbitrary sets, without looking at structure - so they don't consider an ordering structure either.

There is no inherent order in the set of bijections either.

I agree with you.

Thats why I say that order is not important in this case.

 

[ddt]

I agree with you.

Thats why I say that order is not important in this case.

Then why do you bring up the word "order" in the first place when it serves no function at all?

 

[doronshadmi]

The guests are immaterial to the point. The point is that I construct infinitely many non-bijections from N to itself.

Good.

So do you agree that any non-finite set that is mapped to its own members can be in at least 3 different mappings, which are:

Case 1: A bijection between S and some proper subset of S.

Case 2: A bijection of S to itself.

Case 3: A non-bijection of S to itself.

(If you wish you can combine cases 1 and 2 into a one case)

 

[nathan]

At a mapping of S to itself there are no new rooms and no new guests, so Hilbert's analogy is not a mapping of S to itself.

Sigh, you really don't understand mapping functions, do you?

Here's one of Hilbert's mappings, using the notation you introduced in the OP.

  <--> 1
1 <--> 2
  <--> 3
2 <--> 4
  <--> 5
3 <--> 6
  <--> 7
…

After clarification, you claimed the second mapping in the OP is a non-bijective mapping. Here's a non-bijective mapping. Why is this second one of Hilbert's not a mapping of S onto itself?

 

[doronshadmi]

Then why do you bring up the word "order" in the first place when it serves no function at all?

In order to avoid any confusion by people that are not aware of the difference between cardinal and ordinal.

 

[doronshadmi]

Sigh, you really don't understand mapping functions, do you?

Here's one of Hilbert's mappings, using the notation you introduced in the OP.

  <--> 1
1 <--> 2
  <--> 3
2 <--> 4
  <--> 5
3 <--> 6
  <--> 7
…

After clarification, you claimed the second mapping in the OP is a non-bijective mapping. Here's a non-bijective mapping. Why is this second one of Hilbert's not a mapping of S onto itself?

Do you see any new guests (objects) or rooms (objects) here (I don't)?

Do you see a 1-to-1 and onto of S to itself at your example (I don't)?

Where is the mathematical work that does not ignore the non-bijection of S to itself?

 

[nathan]

To clarify my understanding of Doron's claims, I'm paraphrasing them here.

Consider a non-finite set S.

It is presumed that non-finite and infinite are synonyms.

There are infinitely many bijective mappings from S to S and infinitely many non-bijective mappings from S to S.

Agreed.

Standard Math's discounts the non-bijective mappings

I'm not sure what that means. That Standard Maths claims they don't exist? (false), That Standard Maths doesn't make use of them? (false). Something else?

The mappings used in Hilberts hotel..

which standard maths claims are non-bijective mappings of the natural numbers

... are not non-bijective mappings of a set onto itself.

Disagree.

 

[ddt]

The guests are immaterial to the point. The point is that I construct infinitely many non-bijections from N to itself.

Good.

Great! For a change, you see it too.

So do you agree that any non-finite set that is mapped to its own members can be in at least 3 different mappings, which are:

And then you go on to again torture math. The fact whether a mapping is a bijection or not is a property of the mapping, not of the set. You make it sound like it's the property of the set. And we've seen already that for the first example that comes up of an infinite set - the natural numbers - both bijective and non-bijective functions exist.

Case 1: A bijection between S and some proper subset of S.

Case 2: A bijection of S to itself.

Case 3: A non-bijection of S to itself.

Case 1 is called injective or 1-to-1. When you first say a mapping (function, total function) f has domain S and range (codomain) S, and then it turns out it only maps to elements of T, which is a proper subset of S, it doesn't automagically change the range of f to T.

Functions that are injective but not surjective (onto) - your case 1 - are automatically not bijective. So I really don't see the point to separate case 1 and case 3. Case 1 is a subcase of case 3.

 

[ddt]

In order to avoid any confusion by people that are not aware of the difference between cardinal and ordinal.

Until now, nobody brought up the words cardinal and ordinal, and they're immaterial to the discussion at hand. And if there's anyone in this discussion of whom I'm afraid s/he mixes up those...

 

[nathan]

Do you see any new guests (objects) or rooms (objects) here (I don't)?

As I, and ddt, said earlier, there are no new rooms -- that's the whole point! There are only rooms which contain a guest, and rooms which do not contain a guest. The mapping I provided shows how you may move guests between rooms of an initially full hotel, such that rooms become empty and therefore new guests can be accommodated.

Do you see a 1-to-1 and onto of S to itself at your example (I don't)?

Good, this isn't one, it's non-bijective. You asked for a non-bijective mapping, I provided one. Why are you complaining it lacks features of a bijective mapping?

Where is the mathematical work that does not ignore the non-bijection of S to itself?

Hilbert's Hotel. Have you not read and understood ddt's posts?