How Does Complementary Logic Redefine Mathematical Infinity?

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The discussion centers on the concept of Complementary Logic (CL) and its potential to redefine mathematical infinity. Critics argue that the proponent of CL fails to provide a clear logical framework, relying instead on vague assertions about its capabilities. Concerns are raised about the usefulness of a logic system that cannot derive contradictions, as contradictions are essential for evaluating assumptions in traditional logic. The conversation also touches on the relationship between mathematics and real-world applications, emphasizing the need for clarity and rigor in defining terms and concepts. Ultimately, the lack of a concrete definition for CL undermines its proposed advantages over established logical systems.
  • #101
Originally posted by Organic
1) I proved that |P(N)|>=|N| iff |N| is the cardinal of ALL N members.

2) I clime that there is no such a thing the cardinal of infinitely many elements, because they cannot be completed.

Well, you didn't prove 1 assuming N denotes the set of natural numbers, although you assert it with a wrong proof, as you take a fininte cardinal result and put in aleph-0 and claim the answer. Well, here's a counter example:

for all finite cardinals n>n-1 hence aleph-0<aleph-0

and 2. is a defintion! which incidentally you use in part 1. Besides, it doesn't matter what you believe, it matters what you can prove, or disprove. For instance, in what way is N, the set of natural numbers not complete? I mean, there meanings where it is not complete, algebraically. If I list the numbers in N in sequence, which one do I omit? Either give me an example, ro proive I must omit one.
 
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  • #102
Matt,

PROOF:

Let T be the 01 collection with cardinality of P(N).

Let L be the 01 collection with cardinality of N.

T is:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...

In this private case we can see that because T 01 collection is longer then L 01 collection, ...010101 is not in L but it is in T.

We can change the order of T 01 collection and then we shall find another 01 sequence which is not in L but is in T.

In this stage we can clearly say that |T|>|L|.

The width of T 01 collection = |N|

The lengh of T 01 collection = |P(N)|

Because any missing 01 sequence is already in T, we MUST NOT add it to L.

Therefore we can find a 1-1 and onto between 1,2,3,... to any T member.

But because |L|=|N| we can conclude that |L|=|T|.

Now we have |T|>=|L| which is a contradiction.

Therefore transfinite cardinality does not exist.

Q.E.D


This proof holds for finite or infinitely many objects.

Let us see it in a finite 01 collection.


T has the cardinality of P(3)

L has the cardinality of 3


T 01 collection is:

000
111
001
110
010
101
011
100

01 sequence of 101 is already in T therefore we MUST NOT add it to T.

Therefore there is 1-1 and onto between any 1,2,3,... to any T member:

000 <--> 1
111 <--> 2
001 <--> 3
110 <--> 4
010 <--> 5
101 <--> 6
011 <--> 7
100 <--> 8
 
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  • #103
Let T be the 01 collection with cardinality of P(N).

Let L be the 01 collection with cardinality of N.

T is:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...

How do you justify writing T as a list?
 
  • #104
Hi Hurkyl,

Please read this:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

As you can see I can combine between:

...000000
...000001
...000010
...000011
...
...
(...111,...000]

and

...111111
...111110
...111101
...111100
...
...
[...111,...000)

and get T collection:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...
 
  • #105
I must echo Hurkly here - you are, by writing it as a list and applying the diagonal argument to it presuming it is enumerable AND contains all the 'combinations'; you prove this is not possible. END OF PROOF. But no, you carry on...


More abuses of maths are:

1. The set with cardinality |P(N)|? Don't you just mean P(N)? There are lots of sets with cardinality |P(N)|, you can't talk of THE set...

2. Same as above but with N not P(N) in there

3. you don't know what's in L because you've not said what L is


What is the bijection you claim from T to N? Please say it's not by the numbering of the row.


The finite case does not imply the infinite case. You are attempting to say that becuase, in the finite case we only look at the top nxn square in an nx2^n array, we can do the same in the infinite case - but the construction doesn'tdo this. THere are as many rows as columns in the infinite case, and no row is not used in the diagonal argument.
 
  • #106
Let T be the 01 collection with cardinality of P(N).

As you can see I can combine between:

...000000
...000001
...000010
...000011
...
...
(...111,...000]

and

...111111
...111110
...111101
...111100
...
...
[...111,...000)

and get T collection:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...

Why do you think this list has cardinality P(N)?
 
  • #107
Originally posted by Organic
Hi Hurkyl,

Please read this:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

As you can see I can combine between:

...000000
...000001
...000010
...000011
...
...
(...111,...000]

and

...111111
...111110
...111101
...111100
...
...
[...111,...000)

and get T collection:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...


The first collection gives you those with finitely many 1s in, the second gives you the collection with finitelym any 0s. THey both have card aleph-0 so does their union. But it isn't all of the things in T, only the finite and cofinite ones. At no point is the repetitive string...010101010101 in there.
 
  • #108
Because T is a collection of 2^Omega unique memebrs, where Omega
is the result of using the ZF axiom of infinity bulit-in induction on the power_value
of 2^0, 2^1, 2^3, ...
 
  • #109
Because T is a collection of 2^Omega unique memebrs

Why do you think that? What is omega? Is it cardinality of the natural numbers in ZF?

Omega
is the result of using the ZF axiom of infinity bulit-in induction on the power_value
of 2^0, 2^1, 2^3, ...

This sentence makes no sense.

I assume you mean "Omega is the result of applying induction to the sequence 2^0, 2^1, 2^3, ..."

But this makes no sense because this is not even close to the form of something to which one applies induction... induction is, essentially, when you compute values of something based on previous values (when the domain of the function is well-ordered)

Here, each value is given explicitly; 2^0, 2^1, 2^3, et cetera. There is no computation based on previous values.
 
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  • #110
Matt,

Yes ...01010101 is in T but not in L.
 
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  • #111
Hurkyl,

Omega: http://www.mtnmath.com/book/node53.html

Please look at the axiom of infinity.
 
  • #112
Originally posted by Organic
Because T is a collection of 2^Omega unique memebrs, where Omega
is the result of using the ZF axiom of infinity bulit-in induction on the power_value
of 2^0, 2^1, 2^3, ...

As we established last night, you've not defined 'the axiom of infinity built-in induction'. in fact a quick google for the phrase reveals you're the only person on the entire web as google sees it that has used that phrase. There is the axiom of infinity which merely asserts our set theory has some set that contains the Natural numbers (we do: the natural numbers). There is also the principle of induction, which you don't use.
 
  • #113
Originally posted by Organic
Matt,

Yes ...01010101 is in T but not in L.

But you just told us how to construct T, and the ...010101 is not there. You said to add together the finite and cofinite lists to produce the T collection.


And you've still not adequately define L (or T really), but I think we know what they are. Want to make sure?
 
  • #114
No Matt,

The pruduct of using ZF axiom on infinity on the power_value of
2^power_value is an ordered collection of 2^aleph0 unique members.
 
  • #115
Matt,

I did not add anything to T I just changed the order of its members
to show you that your "proof" by left endless zeros does not hold.
 
  • #116
Originally posted by Organic
No Matt,

The pruduct of using ZF axiom on infinity on the power_value of
2^power_value is an ordered collection of 2^aleph0 unique members.

Erm, that's not what the axiom of infinity states. I see you've dropped the word induction.

The axiom assures us that there is a set conatining the natural numbers. It does not allow us to whack in an alaph-0 at will. Aleph-0isn't even part of ZF's axioms.


So, explain what you are doing to
2^0, 2^1, 2^2...

the nearest I can get is that you are taking unions of sets of size 1,2,4,8,...

the union clearly is countable.
 
  • #117
And anyways, induction doesn't require the axiom of infinity, or even much set theory. Here are some definitions:

Definition: < is a well ordering on a set S iff subsets of S have smallest elements (with respect to <)

Notation: We often abbreviate this by saying "S is a well ordered set" and leave the ordering, <, implicit.


Theorem (induction): If

(1) S is a well ordered set with smallest element s
(2) P(x) is a logical proposition
(3) P(s)
(4) (y in S, y < x, P(y)) &rarr; P(x)

Then

z in S &rarr; P(z)


This barely requires any set theory, and doesn't even require things like "for all" or "there exists"
 
  • #118
Originally posted by Organic
Matt,

I did not add anything to T I just changed the order of its members
to show you that your "proof" by left endless zeros does not hold.


from your original definition in the first new diagonal argument my proof is perfectly valid.


I can now prove that any string has either no 1's after some point, OR it has no 0's given your new way of constructing T by interlacing two clearly countable sets, since it came from one of two original lists where one or the other holds.
 
  • #119
Hurkyl,

It is clear as a middle-noon sun, if n exist then n+1 exist.

How can you say that this is not an induction, which its product is clearly Omega(=aleph0)?

Please look here: http://us.metamath.org/mpegif/aleph0.html
 
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  • #120
Matt,

You still don't get it, what you call two lists is the same list with
3 different orders.


The first is top --> bot.

The second is bot. --> top

The third is one form top, one from bot., one from top, one from bot., ...

In all of the cases we have the same 2^aleph0 01 unique members.
 
  • #121
It is clear as a middle-noon sun, if n exist then n+1 exist.

How can you say that this is not an induction, which it product is clearly Omega(=aleph 0)?

What does this have to do with

is the result of using the ZF axiom of infinity bulit-in induction on the power_value
of 2^0, 2^1, 2^3, ...

?


(P.S. I'm content with simply saying &omega; = N)
 
  • #123
Originally posted by Organic
Matt,

You still don't get it, what you call two lists is the same list with
3 different orders.


The first is top --> bot.

The second is bot. --> top

The third is one form top, one from bot., one from top, one from bot., ...

In all of the cases we have the same 2^aleph0 01 unique members.

Priceless! So, your infinite list has two ends?

What is clear is that reading the list



...00000
...00001
...00010
...00111
...01100


that every string on the list has only finitely many elements that are non-zero. My proof holds here: let x be the string on row r, as 2^r>r for all r, it follows that after the r'th postion on the string (right to left) that every subsequent entry is zero as by definition the s'th coumn starts with 2^s zeroes.



Now the second list reading the orignal list from bottom to top (which now implies that the list is finite)

...1111
...1110
...1001
...1000


the same proof demonstrates that any string on the list only has a finite number of 0's in it. Now you are saying these lists are the same? Where is the string ...1111 on th first list? I only ask because in you new diagonal paper you say it isn't on the first list. But it is on the second, and the lists emnumerate the same elements,. namely the thing you call T?




And you say if n then n+1

n is not a statement that is true or false. It is evident you don't understand the slightest thing abuot logic and its conventions. What does it mean for n to be true?
 
  • #124
Theorem:

If S is a set, and subsets of S have both smallset and largest elements, then S is finite.


As a special case:

If L is a list, and each sublist of L has both a first and a last entry, then L is a finite list.
 
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  • #125
Matt,

My proof is not about |T|=|L| but |T|>=|L|.

in T there are:

...000...1111
...000...1110
...000...1001
...000...1000

and also there are:

...111...1111
...111...1110
...111...1001
...111...1000

You still don't understand that there is no such a thing like cardinality of infinitly many objects, because any collection of infinitly many objecsts is
a non-complete collection.

Therefore its cardinality cannot be found.
 
  • #126
Not every collection can be represented by a list.
 
  • #127
Originally posted by Organic
Matt,

My proof is not about |T|=|L| but |T|>=|L|.

in T there are:

...000...1111
...000...1110
...000...1001
...000...1000

and also there are:

...111...1111
...111...1110
...111...1001
...111...1000

You still don't understand that there is no such a thing like cardinality of infinitly many objects, because any collection of infinitly many objecsts is
a non-complete collection.

Therefore its cardinality cannot be found.

The wholepoint of this is that you ARE claiming |T|=|L|, how do you do this? by claiming the list for T has cardinality 2^aleph-0, that it is a complete list. It isn't as you know because of th diagonal argument.

Explain why N is not complete, in your view; cardinality does make sense for infinite sets; answer any of the fatal errors in your posts today.
 
  • #128
Hurkyl,

I am talking about (...111,...000] XOR [...111,...000)
XOR (...101010,...111] XOR (...010101,...000] XOR ...
 
  • #129
Matt,

The fatal error is the cardinality of infinitly many elements.


Please show me how can infinitely mant elements can be complete.
 
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  • #130
(...111,...000] XOR [...111,...000)
XOR (...101010,...111] XOR (...010101,...000] XOR ...

(a) We've told you over and over again that this notation is nonsensical; it doesn't even come close to the standard mathematical definition, and you refuse to define it (and consistently claim that you are using it in a standard way)

(b) You're trying to make a list. What does exclusive or have to do with anything?

(c) You do realize that if A xor B xor C xor ... does NOT mean "Exactly one of A, B, C, ... are true"; it means "An odd number of the A, B, C, ... are true".

e.g.: if A, B, and C are all true, then A xor B xor C is true.
 
  • #131
Originally posted by Organic
Hurkyl,

I am talking about (...111,...000] XOR [...111,...000)
XOR (...101010,...111] XOR (...010101,...000] XOR ...


Oh, look, more misuse of logic. XOR is a logical predicate, its inputs are things that are eithert true or false. what does it mean of (...1111,...000] to be true. Hell, for that matter, what does it mean? I guess you'll post about complementary logic, even though it isn't clear why you'd do that because all the questions are about proper mathematical objects.
 
  • #132
Please show me how can infinitely mant elements can be complete.

(I'm guessing at what you mean by complete)


If C is a collection, then C = C, right?

Then, C is clearly a complete collection of the elements of C.

N is clearly a complete collection of the elements of N.

R is clearly a complete collection of the elements of R.
 
  • #133
Ok Hurkyl,



I am talking about (...111,...000] OR [...111,...000)
OR (...101010,...111] OR (...010101,...000] OR ...
 
  • #134
A or B or C or ... means "At least one of the A, B, C, ... are true"

A and B and C and ... means "All of the A, B, C, ... are true"


Oh, I should also note that logical expressions may only have a finite number of terms.


That still doesn't address the issue that

(...111,...000]

has not been given a meaning comprehensible to anyone but yourself.


And it still doesn't address the issue that this seems to have absolutely nothing to do with lists. Based on your earlier posts, I might guess you're trying to say "Do something that corresponds to (...111,...000], then the something that corresponds to [...111,...000), then ..."

But I know neither what (...111,...000] is, nor what that something corresponding to (...111,...000] is.
 
  • #135
Originally posted by Organic
Matt,

The fatal error is the cardinality of infinitly many elements.


Please show me how can infinitely mant elements can be complete.


Let N be the collection of all natural numbers, it is complete in the sense that it contains all the natural numbers, and its existence is not contrary to the ZF axioms (in fact it is required by the axiom of infinity). It is a set.


Let Z be the ring defined by formally adding inverses to elements in Z and including 0. Clearly it is still a set, and is 'complete' in any reasonable sense - that is the opereations of addition and subtraction do not take one out of the set. It doesn't contain any pink elephants, but then its existence doesn't require pink elephants.


Form the fraction field, clearly this is still a set as the construction of itself proves (there are countably many elements in any equivalence class, and countably many equivalence classes. It is Q, and it is again complete in terms of algebraic operations..

perhaps you should define what you mean by complete?




Let Q be
 
  • #136
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.

When infinitely many elements are forced to be notated as one concept
we are dealing with actual infinity, and no theory can use actual infinity as input, and cannot explore it.

Please see: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf
 
  • #137
Originally posted by Organic
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.

When infinitely many elements are forced to be notated as one concept
we are dealing with actual infinity, and no theory can use actual infinity as input, and cannot explore it.

Please see: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf


So it is your personal meaning for complete that no one else understands? What quantity do you mean?


Anyway, why haven't you answered the questions posed to you?

Does this mean you accept that your attempts to show |T|=|L| is non-sense?
 
  • #139
Looking back through your posts, it seems that you mean

a set C, is not complete, if given any list (enumeration) of some (possibly all) of its elements, then there is some element of C not listed.


This is exactly the defintion of uncountable.

N is 'complete' as the trivial enumeration shows, so complete is countable? Since N exists countable/complete sets exist, and clearly you accept Cantor's diagonal argument, hence there aer uncountable sets. Which in your world would be 'incomplete'. As there already exists a word 'countable' and 'complete' hasn't been well defined by you, can I suggest this as a definition, and simultaneous ask that you stop using it 'cos complete is used in lots of situations already, and why invent another name for something already known
 
  • #140
Originally posted by Organic
Matt,

Please read again pages 3 an 4 in:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

and also look again at:

http://www.geocities.com/complementarytheory/Identity.pdf

Thank you.


Why, what are these going to answer?

Just provide a defintion of complete here, shoulc be a matter of a few lines. We don't need to reread new diagonal because there are aleraedy enough errors explained to you here that you've failed to address, why would we add more.
 
  • #141
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.
 
  • #142
Hurkyl,

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

Thank you.
 
  • #143
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
 
  • #144
For me "complete" can be used only for a finite collection, where all memebers are unique (can be clearly distinguished from each other).
 
  • #145
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?
 
  • #146
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.
 
  • #147
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.
 
  • #148
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?
 
  • #149
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?
 
  • #150
A definition for complete:

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

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