How Does Complementary Logic Redefine Mathematical Infinity?

[Hurkyl]

If a set has infinitely many elements we cannot use the word "complete" from a quantitative point of view, because no quantity can be captured and notated by one symbol, and then can be used as a meaningful input for some mathematical system.

What does "complete from a quantitative point of view" mean?

Why not?


Anyways, one thing I've told other people who don't think cardinality appropriately captures the idea of quantity is:

Don't think of cardinality as appropriately capturing the idea of quantity.

Cardinality has a rigorous set theoretical definition, which is not "Cardinality is the size of a set."

So treat it as such. It is yet another abstract mathematical idea that mathematicians use because it happens to be useful.

The same is true about ordinality. If you don't like the idea of counting to infinity and beyond, then treat it as it really is; a useful, abstract mathematical idea.

The only reason any mathematician would think as cardinal numbers as a size or ordinal numbers as counting is because it helps the mathematician understand things. For example, for me personally, such an interpretation has given be a very good intuition about transfinite induction, allowing me to very naturally extend proofs that apply in countable cases to proofs that work in uncountable cases. (Such as the proof that every vector space has a basis)

But if you don't like to think of cardinality as size and ordinal numbers as counting numbers, then don't, because, in all technicality, cardinality is not size, and ordinal numbers are not counting numbers.

 

[Organic]

Hurkyl,

Can you say in simple English what is Cardinality and Ordinality to you?

Thank you.

 

[matt grime]

Nice idea, Hurkyl. Here's how I might do it:

for every set, S, define a symbol #S, say that #S~#T iff there is a bijection from S to T. Call the equivalence classes of this relation cardinals.

For fininte sets, we can simply set #S to be the number of elements in S, for infinite sets we pick some distinguished labels

 

[Organic]

For me "complete" can be used only for a finite collection, where all memebers are unique (can be clearly distinguished from each other).

 

[Organic]

Matt,

If #S has some unique property (not quantitative) and #T has some unique property (also not quantitative) how do you difine a bijection between #S and #T?

 

[matt grime]

Originally posted by Organic
Hurkyl,

Can you say in simple English what is Cardinality and Ordinality to you?

Thank you.

I suspect my answer, would be yes and no. Yes to the satisfaction of a mathematician, but no to you. As this is a mathematical defintion, you lack of understanding isn't important, because you are trying to do maths only on things that have some nice worldly explanation.

Q. What is integration?

A. Naively it is 'the area under the graph' but you can integrate things that don't have graphs. You might even cite anti-derivatives, but that is only true again in certain circumstances (when the integrand is a continuous function for instance).


THings in maths are what they do, they are their definitions. Nothing more nothing less. For simple things there might be some real world explanation. Maths would be a lot better off if this attitude were discouraged. However, arguably its students at a basic level would be worse off.

 

[matt grime]

Originally posted by Organic
Matt,

If #S has some unique property (not quantitative) and #T has some unique property (also not quantitative) how do you difine a bijection between #S and #T?

I'm not defining a bijection between #S and #T. They aren't sets, so I don't define functions between them.

If we presume you mean S and T as sets, it is often very hard to decide if there is a bijection between them. That difficulty doesn't stop me saying that #S~#T if there is a bijection from S and T. Computability has nothing to do with it. The bijection is purely a set theoretic statement.

The even perm group on 4 elements is not isomoprhic (as a group) to the symmetries of a hexagon, but as sets they have the same cardinality.


You say 'complete' is only meaningful for finite sets; you've not still defined complete.

 

[Organic]

Matt,

You use a lot the word complete, so as a mathematician you have a definition for it, so please tell me what is complete from mathematician point of view?

 

[matt grime]

Here I woulduse complete to mean contains all the things that it ought to, all the things it can. The usual defintion for complete as we know it.


Something is not complete here if we can demonstrate that there exists something not there that ought to be - exactly as we do in Cantor's proof for the uncountability of R.



THere are mathematical definitions for complete that are contextual - a normed space is complete if every cauchy sequence converges. There may be others but I can't think of them off hand.


If you want to define complete for sets do so. So far your only attempt is to say complete is BY DEFINITION finite. Which is a bad one.

Figured out any of the answers to the questions asked?

 

[Organic]

A definition for complete:

A property that depends on the existence of all its elements in one and only one collection.

 

[matt grime]

That isn't a definition of complete. First it doesn't say what things might be complete.

It should start: A set S is complete iff then some criteria


you've just said complete is a property, but not what tha property is. As every set S contains all its elements trivially every set has this notional complete property. And seeing as every set with at least two elements is the union of two non-empty sets, you are also being inconsistet by demanding 'only one'


So you now admit that all this time you've been arguing with Hurkly and me, you've never known what it was that you were arguing about? Right, I'm off, that really is the final straw. To be accused of not understanding your ideas when you now admit you don't know what they are...? You don't think that a little too much?

 

[Organic]

Matt,

Your definition is: "contains all the things that it ought to"

Is this better then?

"A property that depends on the existence of all its elements in one and only one collection."

 

[Organic]

Also By my definition we can clearly understand why infinitely many elements cannot be completed.

By your definiton N is complete.

By my definition N is not complete because there is no such a thing all infinitely many elements.


I have to go so, bye bye for today.

 

[matt grime]

Oh dear I seem to be trapped.

My working defintion for complete was one that was a best attempt to put *some* mathematical meaning there to answer your challenge to give an infinite complete set. I don't claim that that is a mathematical definition. I wouldn't chose to use complete in this sense as a mathematical defintion. And as you didn't give any definition it was the best I could come up with.

If you were to press me, I would say that in the case we are talking about, some list L of elements of a set S is a 'complete' list if it enumerates all the elements of S, that is gives a bijection between N (or some subset), the natural numbers, and S.

But I still can't decide what complete might mean for a set.

Personally I would not choose to use the word complete like this in a rigorous mathemmatical context, except to give it its usual meaning in the English language - which is what my 'contains all that it ought to' was.The best off the top of my head definition of complete as an adjective - not lacking any components.


I don't see how a set cannot be complete in this sense. What element of the natural numbers is missing from N?

And, yes I would say it was better than yours as it says what complete means, yours just says complete is a property of some objects that are ambiguously defined.

 

[matt grime]

Originally posted by Organic
Also By my definition we can clearly understand why infinitely many elements cannot be completed.

By your definiton N is complete.

By my definition N is not complete because there is no such a thing all infinitely many elements.


I have to go so, bye bye for today.

But you've not defined complete!

"A property that depends on the existence of all its elements in one and only one collection"

does not define complete.

Continuity depends on the existence of an epsilon, but that doesn't say what continuous means, does it?

So, simply rephrase that,

A set is complete if and only if... WHAT?!?

it has all its elements in one and only one set?

Obviously garabage.


The set N has all its elements in the set (collection!) N, so why isn't that complete, whatever the hell that means.

"no such a thing all infinitely many elements"


EH? so all sets are finite?


I've never met anyone as obstinate about refusing to understand the bleedin' obvious. Have a medal.


I notice you are still refusing to answer any of the criticisms of your argument.

 

[matt grime]

Summary of the arguments so far:

The following terms have been introduced by you as mathematical terms without definiton despite being asked repeatedly:

opposite; non-linearity; complementary; symmetry-degree; fading transition; structural quantitative; information point; uncertainty; redundancy; {__}; approach; closeness; mutation.

We also have the axiom of infinity induction that is a mystery to everyone as you are the only person to have ever used this phrase (except for those asking what it means).
Then there are tautologies that are false; logical operations of 'and' etc 'defined' on inputs that are not statements that are true of false; the claim that an inequality is EQUAL to a set.
Not to mention this long argument about what complete means in the rigorous sense that you can't define, but use freely. Your *opinions* about it don't count for anything mathematically, it is only what you can prove from the existing mathematics that counts in this argument, you must use the existing definitions and conventions, you cannot attempt to claim it is inconsistent if the inconsistencies you claim are not based on statements within the system. You cannot say that current mathematical thinking is flawed because it does not agree with something you've just made up on the spot without knowing anything about the maths you claim to be talking about.
Fine, if you want to claim it doesn't do what you want then add to it, use some other set theory (ZF v ZFC) but don't tell us is wrong internally, when the only way you can do so is to use external objects.

You did define entropy as a partition of a number into equal parts.

And we've not sorted out what it is that you mean by cardinality not being defined for infinite sets. It is, at best you could argue that you think all infinite sets have the same cardinality. You've proved that to be false.
And then there is the misuse of lists...

 

[Hurkyl]

Can you say in simple English what is Cardinality and Ordinality to you?

Sure. I intuit Cardinality as being (a generalization of) the size of a set, and Ordinality to be (a generalization of) counting numbers.


When it comes down to rigorous work, though, I take cardinality to be something that has the property that:

Set A has cardinality no less than set B iff there is a function from A onto B. (written |A| >= |B| or |B| <= |A|, also said as "set A has cardinality at least as great as set B", or a variety of other ways)

Sets A and B have the same cardinality (written |A| = |B|) if |A| <= |B| and |B| <= |A|.

This is an equivalence relation, so we can form equivalence classes, and the things we use to represent these classes we choose to call cardinal numbers.


I'll get to ordinality later.

 

[Hurkyl]

Lemme take another angle at pointing things out...

Cantor's diagonal argument requires there to be a map from the set of columns onto the set of rows. (Actually, the index set for the columns and the rows is supposed to be the same set, but you can tweak the argument a bit to make it a bit more powerful)

(If you want, I can go through the argument and point out exactly where this is required)

e.g.

If we have a list that is 3 wide, and 3 tall, then there is indeed map from the set of columns (of size 3) onto the set of rows (of size 3), so Cantor's diagonal argument is guaranteed to give a sequence not in the list.

However, if the list is 3 wide and 4 tall, no such onto function exists. Thus, Cantor's diagonal argument is no longer guaranteed to give a sequence not in the list. (though it may get lucky!)


You don't account for this when you try to apply Cantor's diagonal argument to the list of all binary sequences. The columns are indexed by N, and the rows are indexed by P(N). Until you can prove that there is a function from N onto P(N), it is fallacious to apply Cantor's diagonal argument to this list.


I've kind of lost track of what you're trying to assert these days; if you're still trying to claim |N| = |P(N)|, then your argument is circular, because you're using Cantor's diagonal argument to do your proof, but Cantor's diagonal argument can't be used until you've proved it!

 

[Organic]

Matt,

Let us start form this definition.

A set is complete if and only if all its objects can be found.

exapmle: S={1,2,3,...} is not a complete set.

 

[matt grime]

Originally posted by Organic
Matt,

Let us start form this definition.

A set is complete if and only if all its objects can be found.

exapmle: S={1,2,3,...} is not a complete set.

Ah, but what do you mean by found? You give me any natural number and I bet I can 'find' it on that list, you see it's the order in which things are given, you take it to mean that the list must terminate, I take it to mean that given the list and any number (which is then fixed and finite call it n) that I can find it in approximately n steps - look at the n'th term. It changes as n changes but so what?

Your hand wavy arguments do not show that the set of natural numbers and its power set have the same cardinality as you want to claim. That claim is errant nonsense and shows you don't understand things you claim to use accurately. So what that there is no finite time algorithm for listing ALL the natural numbers, that isn't important. As with so many cranks you're confusing what can be constructed from what can be defined.


Just associating some 'property' to something does not a priori contradict anything, you would need to show that this 'property' meant that constructions used (in the Cantor argument here) would fail to work. But they don't; they DO work. Just because you don't understand the convention
that it is possible to specify an infinite set or string doesn't make you right.

 

[Organic]

One by One is not all.

 

[Organic]

A set is complete if and only if all its objects can be found.

It means that S have first an last objects.

exapmle: S={1,2,3,...} is not a complete set.

 

[matt grime]

Eh? You've just given me a list of the Natural numbers and challenged me to 'find' all the natural numbers. I can 'find' them simply by saying there they are, that set there you've just written down, by YOUR definition contains all the natural numbers. My usual challenge stands, tell me which natural number is not on that list, go on, just one. Or define 'find' better.

I mean, I could ask you to prove in your next post that you're not the Dalai Lama, and even tell you I@ve a simple rule that if you write X, I'll believe, you, only I won't tell you what X is, and nothing else will suffice.

You're plumbing new depths here, even by your standards.

 

[matt grime]

Originally posted by Organic
A set is complete if and only if all its objects can be found.

It means that S have first an last objects.

exapmle: S={1,2,3,...} is not a complete set.

So we conclude that a set is complete if and only if it is finite. AND?

 

[matt grime]

incidentally, what if the notion of first and last is non-sensical? I mean is the openinterval (0,1) complete in your opinion?

 

[Organic]

No the open interval (0,1) is not complete.

 

[matt grime]

What about [0,1] the closed interval, that has a first and last element in some sense.

 

[Organic]

In this case [0,1] is not (0,1)+ 0 and 1, but some finite collection between 0 and 1, + 0 and 1 included.

 

[matt grime]

Originally posted by Organic
In this case [0,1] is not (0,1)+0 and 1 but some finite collection between 0 AND 1, + 0 AND 1 included.

Good, you're at last beginning to understand things you use.

So, a set is complete iff it is finite.

Now kindly explain what the hell that has to do with Cantor's Argument?

 

[Organic]

Matt,

Now kindly explain what the hell that has to do with Cantor's Argument?

I answer only to polite persons.

 

[matt grime]

This is mathematics, not a personality contest, if you think me rude AND wrong, then what better way to correct me than to demonstrate why it is that because N is not finite that Cantor's argument is wrong.

Note, 1. Youve proved it is correct, so this is going to be a test of your consistency in mathematics.

Note, 2. I imagine you're about to wheel out the 'it makes no sense to use the word 'all' for things that aren't complete [finite]'


For 2. Why is it not permissible to say N is the set of all natural numbers? I mean if it's only because the set is not finite then I maintain, as would any mathematician, that you are not using 'all' in the context of that sentence correctly. The inequality n>n-1 is clearly true for ALL natural numbers.

N is be definition the set of ALL natural numbers. It contains all its elements. The way to show 'not all' is to find one counter example, so show me some natural number not in the set of all natural numbers.

We can talk about the set of all natural numbers, the set of all complex numbers... we cannot talk about the set of all sets obviously.

 

[Organic]

Dear Hurkyl,

x=2

Let us examine x and P(x)

x:

21
^^
||
vv
01<-->1
10<-->2

P(x):

 1 0
2 2
^ ^
| |
v v
0 0 <--> 1
0 1 <--> 2
1 0 <--> 3
1 1 <--> 4

====================================================================================================

x=3

Let us examine x and P(x)

x:

321
^^^
|||
vvv
001<-->1
010<-->2
011<-->3

P(x):

 2 1 0
2 2 2
^ ^ ^
| | |
v v v
0 0 0 <--> 1
0 0 1 <--> 2
0 1 0 <--> 3
0 1 1 <--> 4
1 0 0 <--> 5
1 0 1 <--> 6
1 1 0 <--> 7
1 1 1 <--> 8

====================================================================================================


x=|N|

Let us examine x and P(x)

x:

   4321
   ^^^^
   ||||
   vvvv
...0001 <--> 1
...0010 <--> 2
...0011 <--> 3
...0100 <--> 4
...

If we use the ZF axiom of infinity induction on the power_value
of each column then:

P(x):

 [b]   3 2 1 0[/b]
   2 2 2 2
   ^ ^ ^ ^
   | | | |
   v v v v
...0 0 0 0 <--> 1
...0 0 0 1 <--> 2
...0 0 1 0 <--> 3
...0 0 1 1 <--> 4
...0 1 0 0 <--> 5
...0 1 0 1 <--> 6
...0 1 1 0 <--> 7
...0 1 1 1 <--> 8
...1 0 0 0 <--> 9
...1 0 0 1 <--> 10
...1 0 1 0 <--> 11
...1 0 1 1 <--> 12
...1 1 0 0 <--> 13
...1 1 0 1 <--> 14
...1 1 1 0 <--> 15
...1 1 1 1 <--> 16
...

As we can see, we get an ordered collection of infinitely many 01 unique sequences with Width aleph0 and Length 2^aleph0.


Conclusions:

1) Both Width and Length are enumerable.

2) Length > Width.


Cantor's diagonal method problems:

1) It was examined on arbitrary unordered collection of 01 sequences.

2) No collection of 2^aleph0 was examined.

3) Any "missing" 01 sequence is already in the 2^aleph0 collection,
therefore it MUST NOT be added to the collection.

4) Any complete collection cannot be but a finite collection.

5) Infinitely many elements cannot be completed, therefore the cardinality of infinitely many elements is unknown and cannot be used to establish the transfinite system.

 

[matt grime]

Edited:


Ah, so you have stopped answering me.

Anyway, you're claiming 'the axiom of infinity induction' again.

Still you are the only person to have done this, what is it?


Clearly the object you construct has both countable rows and columns because you list them, and, ergo, they are enumerable.

At least you are consistent in your nonsense, shame you really don't understand the elementary explanations given to you as to why what you're claiming is wrong.

As is eminently clear every row in your array is eventually zero by construction, so how can it possibly have cardinality 2^aleph-0. In fact you are ASSUMING alpeh-0 equals 2^aleph-0 to prove they are equal and derive a contradiction... Some one might think you didn't have the slightest idea about mathematics. They'd be right seeing as for the last God know's how long you've been insisting that 'no infinite set is complete'. In post 34 in this thread approx. you were asked to define that or at least explain what you meant. For the next 130 posts you were unable to provide any answer to that. Eventually after much prompting we were able to decide 'complete' means finite!

Now, how do we get you to see this claim of yours is wrong?

 

[Organic]

Matt,

Please read carefully what I write here.

You wrote:

This is mathematics, not a personality contest

1) For me the quality of communication between persons comes before any subject.

2) By "the quality of communication" I mean that no person in this process, using any aggressive or fanatic approach related to the examined subject.

3) I think that fruitful communication is based on cool head ,open mind, and positive approach that first of all tries to understand the other person.

4) If you can accept these basic terms of "the quality of communication" than please write your posts without “wtf”, “what the hell”, ”utter b***”, and I’ll be glad to communicate.


Yours,

Organic

 

[matt grime]

Fanatic? You are not fanatically pushing a completely stupid idea? Sorry, Organic, you got the polite 'please elaborate' posts. You didn't, just obfuscated and said that Hurkly and I didn't understand anything and yet you couldn't even post a defintion of 'complete' for 130 posts, and even then only with severe prompting and a lot of help.

Is it any wonder you've managed to frustrate me into these outbursts? Some people have just got fed up with even dealing with you nonsensical 'proofs'.

So, a set is complete iff finite, what has that to do with Cantor's argument that you find so wrong? Rememeber it's the maths that counts, not your opinion of what should and shouldn't be done with infinite sets.