How Does Complementary Logic Redefine Mathematical Infinity?

[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 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 ω = N)

 

[Organic]

Hurkyl,

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

And see by yourself the connection.

 

[matt grime]

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?

 

[Hurkyl]

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.

 

[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.

 

[Hurkyl]

Not every collection can be represented by a list.

 

[matt grime]

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.

 

[Organic]

Hurkyl,

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

 

[Organic]

Matt,

The fatal error is the cardinality of infinitly many elements.


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

 

[Hurkyl]

(...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.

 

[matt grime]

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.

 

[Hurkyl]

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.

 

[Organic]

Ok Hurkyl,



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

 

[Hurkyl]

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.

 

[matt grime]

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

 

[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

 

[matt grime]

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?

 

[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.

 

[matt grime]

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

 

[matt grime]

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.

 

[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.