Here more details about layer set theory,

my favourite part of layer theory:

The central idea is to treat “x is element of set M” (x e M) as a layer statement:

It is true in layer t+1 that set x is element of the set M, if the statement A(x) is true in layer t.

Equality of layer sets:

W (M1=M2, d+1) = W ( For all t: W(xeM1,t) = W(xeM2,t) , 1 )

Especially: W (M=M, d+1)=w for d>=0.

The empty set 0:

W(x e 0, t+1) := W( W( x e 0, t ) = w , 1 ) = -w for t>=0.

The full set All:

W(x e All, t+1) := W( W( x e All, t ) = w v W( x e All, t ) = u v W( x e All, t ) = -w , 1 ) = w for t>0 and =u for t=0.

So other than in most set theories in layer theory the full set is a normal set.

Axiom M1 (assignment of statements to sets):

W(x e M, t+1) := W ( W ( A(x), t ) =w1 v W ( A(x), t ) =w2 v W ( A(x), t ) =w3 , 1 )

with w1,w2,w3 = w,u,-w

For every layer set M there exists a layer logic statement A(x) witch fulfils for all t=0,1,2, …:

W(x e M, t+1) = W ( W ( A(x), t ) = w v W ( A(x), t ) = -w , 1 )

W(x e M, 0+1) = W ( W ( A(x), 0) = w v W ( A(x), 0 ) = -w , 1 )

= W (u=w v u=-w, 1 ) = -w

Axiom M2 (sets defined by statements):

For every layer logic statement A(x) about a layer set x there exits a layer set M so that for all t=0,1,2,3,… holds:

W(x e M, t+1) := W ( A(x), t ) (or the expressions of axiom M1).

Definition M3 (definition of meta sets):

If F is a logical function (like identity, negation or f.e. FoW(xeM1,t) = W(xeM1,t)=w )

then the following equation defines a meta set M: (M1=M is allowed):

W(x e M, t+1) := W ( F o W(x e M1, t), 1 )

Consequences of the axioms and definitions:

In layer 0 all sets are u:

W( x e M, 0 ) = u (as all statements in layer 0).

In layers > 0:

W(x e M, t+1) := w if W ( A(x), t ) = w else W(x e M, t+1) := -w

For all x and (normal layer) sets M holds: W(x e M, 1) = u (as W(A(x),0)=u).

For all x and meta sets M holds: W(x e M, 1) = w or –w

Last not least let´s look upon the Russell set:

Classic definition: RC is the set of all sets, that do not have themselves as elements

RC:= set of all sets x, with x –e x

In layer theory: W(x e R, t+1) := W ( W ( x e x, t ) = -w v W ( x e x, t ) = u , 1 )

W(x e R, 0+1) = W ( W ( x e x, 0 ) = -w v W ( x e x, 0 ) = u , 1 ) = W (u=-w v u=u , 1 ) = w

Therefore W(R e R,1) = w

W(R e R,2) = W ( W ( R e R, 1 ) = -w v W ( R e R, 1 ) = u , 1 ) = W (w=-w v -w=u , 1 ) = -w

And so W(R e R,3) = w, W(R e R,4) = -w , …

R is a set with different elements in different layers, but that is no problem in layer set theory.

As All, the set of all sets, is a set in layer theory, it is no surprise,

that the diagonalisation of cantor is a problem no more (I just give the main idea):

Be M a set and P(M) its power set and F: M -> P(M) a bijection between them (in layer d)

Then the set A with W(x e A, t+1) = w := if ( W(x e M,t)=w and W(x e F(x),t)=-w )

A is a subset of M and therefore in P(M).

So it exists x0 e M with A=F(x0).

First case: W(x0 e F(x0),t)=w , then W(x0 e A=F(x0), t+1) = -w (no contradiction, as in another layer)

Second case: W(x0 e F(x0),t)= -w then W(x0 e A=F(x0), t+1) = w (no contradiction, as in another layer)

If we have All as M and identity as Bijektion F we get for the set A:

W(x e A, t+1) = w := if ( W(x e All,t)=w and W(x e x),t)=-w ) = if ( W(x e x),t)=-w )

This is the layer Russell set R (I omitted the ´u´-value for simplification)-

and no problem.

So in layer theory we have just one kind of infinity – and no more Cantor´s paradise …

Yours,

Trestone

Hello,

perhaps I should tell a little bit more about the motivation for my axioms:

“Axiom 1: Statements A are entities independent of layers,

but get a truth value only in connection with a layer t,

refered to as W(A,t).”

My introduction of layers to logic was a little bit similar to Max Planck introducing quants in physics:

Not very plausible and I do not really like it, but it seems to work …

With the layers, a liar statement was easier to handle:

If we define statement L by the following:

For all layers t: W(L,t+1) := W (W(L,t) = -w ,1)

“The value of thus statement L is true (in layer t+1) iff the value of L is not true (in layer t”

With the (universal) start W(L,0) = u we get: W(L,1) = -w, W(L,2) = w, W(L,3) = w, W(L,4) = w, …

So L , which is similar to the liar statement, is a statement with alternating truth values and this is allowed and no problem in layer theory thanks to the layers.

The next very special axioms are the axioms about meta statements:

“Axiom 5: (Meta-)statements M about a layer t are constant = w or = -w for all layers d >= 1.

For example M := ´W(-w,3)= -w´, then w=W(M,1)=W(M,2)=W(M,3)=...

(Meta statements are similar to classic statements)

Axiom 6: (Meta-)statements about ´W(A,t)=...´ are constant = w or = -w for all layers d >= 1.”

For the motivation of these axioms, we have a look at axiom 1:

There we find formulations like “for all layers t”, so axiom 1 is a statement about all layers.

As there is a hierarchy of layers and we are only allowed to use smaller layers for defining a truth value

of a statement in a certain layer t0

(I have not formalized this completely yet) the axiom 1 can not belong to a certain layer t1.

On the other side I did not want to have infinite ordinal numbers in my layers,

so I made statements about one ore more layers independent of layers by defining axiom 5.

(I am not sure, if axiom 6 is needed at all, as layers are always in connection with truth values up to now.)

Now we can also handle another liar statement LA:

For all layers t: W(LA,t+1) := W (For all d>0: W(LA,d) = -w ,1)

LA is a meta statement, therefore we can write:

For all layers t: W(LA,t+1) := W (For all d>0: W(LA,1) = -w ,1)

With t=0: W(LA,1) := W (W(LA,1) = -w ,1)

Case 1: W(LA,1)=w then W(LA,1)= W (w = -w ,1) = -w , what is not allowed.

Case 2: W(LA,1)=-w then W(LA,1)= W (-w = -w ,1) = w , what is not allowed.

So LA is not a well defined (meta) statement in layer theory – and therefore no problem.

Similar with liar statement LE:

For all layers t: W(LE,t+1) := W (It exists a layer d0>=0: W(LE,d0+1) = -w ,1)

LE is a meta statement, therefore we can write:

For all layers t: W(LE,t+1) := W (It exists a layer d0>0: W(LE,d0+1) = -w ,1)

With t=d0: W(LE,d0+1) := W (It exists a layer d0>0: W(LE,d0+1) = -w ,1)

Case 1: d0 exists: W(LE,d0+1) := W (It exists a layer d0>0: W(LE,d0+1) = -w ,1) = w, what is not allowed.

Case 2: d0 does not exist: W(LE,d0+1) := W (It exists a layer d0>0: W(LE,d0+1) = -w ,1) = -w, what is not allowed.

So LE is not a well defined (meta) statement in layer theory – and therefore no problem.

I do not know if we have to define (and with what more details), that all statements A need a definition of the form:

“For all layers t: W(A, t+1) = …”

and on the right sides only statements of layers smaller than t+1 or meta statements are allowed.

Yours,

Trestone

Hello,

also layer theory looks very similar to "Russells theory of types" it is different:

For example self reference like "set x is (not) element of set x" is allowed.

(hierarchy and avoiding of self reference is done in another dimension: the layers)

Perhaps someone will risk a second look ...

Yours

Trestone

Hello,

elementary everyday logic should be enough to understand my layer theory.

I myself have studied mathematics and philosophy - but some 20 years ago -

and books and long articles about logic are too boring for me (as would be my thread ...)

To understand my layer theory, I think it is important to understand my motivation, to look for a new kind of logic:

The liar´s paradox always has fascinated me - as a sign that to logic "the jury is still out"

and that in spite of a thousand year old rather stable tradition

we still have not reached full understanding of our ways of thinking

Wikipedia liar´s paradox

My new logic, layer logic, shows a way around this paradox by using (meta) layers (or meta levels).

The key idea of layer logic is, that there is a kind of additional new dimension in logic, the layers.

These layers are hierarchically arranged and have discrete values 0,1,2,3, …

A statement has not a truth value "true" or "false" any more, but a truth value in every layer

and some statements (like the liar statement) have different truth values in different layers.

This is near to the idea and solution of Alfred Tarski (but I do not know details of his proposals and layers/(meta)levels)).

The following two ideas are my own invention (at least I think so):

- Layer 0 (Zero), and the idea that every statement has truth value "u" (=undefined) at layer 0.

- Meta statements, especially the idea that every meta statement has an identical truth value in all layers > 0

(that means it is almost a classic statement).

The liar paradox is connected with some other problems:

Russell´s set; Cantor´s paradox of the power set / diagonalization;

Gödel´s incompleteness theorem, the halting Problem of informatics, EPR and Bell´s theorem in physics

(You can look up this (and other references below) all in Wikipedia for example,

but especially the last three or four problems and their proofs are hard technical stuff –

and it is not necessary to study them for the understanding of layer theory)

If we have a look at the connected proofs to this problems,

we always find a proof by contradiction (connected with the law of the excluded middle (tertium non datur)).

One try to come around the problems with the liar (or the proofs by contradiction)

was multivalued logic, but the extended liar ( "this statement has not the truth value "true" " ) still is a problem there.

Even as I am using a third truth value "u" at layer 0 (It helps to start symmetrically),

I think that this is not the most important part of layer theory.

Another try to change logic and mathematics was intuitionistic or constructive logic/mathematics.

http://en.wikipedia....ki/Intuitionism

Here proofs by contradiction are not valid anymore.

But as this complicated the life of mathematicians a lot, most did not follow this line,

although Gödel´s incompleteness theorem had shaken classical mathematics (like ZFS).

In level theory we would have a third way:

It is (at least in some points) not so radical as constructive mathematics and proofs by contradiction are still possible,

but only within one layer.

As almost all classical proofs to the problems above use multiple layers, they are no longer valid.

The reconstruction of most parts of classical mathematics is possible in layer theory, with some exceptions:

I could not show, that there is only one prime factorisation for every natural numbers over all levels.

The square root of 2 might be rational (with different fractions in different layers).

The nicest part of layer theory in my opinion is the set theory, here we have the set of all sets as a set and only one kind of infinity.

But as I am more a philosopher than a mathematician,

there is still a lot to be controlled and proofed

and a better and complete formalisation is yet to be done (help welcome!).

And if the theory is valid, my next question is,

where and how we can use it to solve problems

(even up to now unsolvable ones and not only in mathematics …)

Yours

Trestone

Hello,

In the layer logic defined in this thread there is an inconsequence:

The axioms 5 and 6 for meta statements, especially that the truth values of meta statements are the same for all layers t>0

are in objection to the principle, that no information about higher (or same) layers is available at a lower level.

I.e. W(W(A,t)=w,1)=w woul allow to conclude in layer 1, that W(A,t)=w even for t>1.

Without axioms 5 and 6 for meta statements it is more complicated to define a value for statements over all layers,

nevertheless I think we should go this more consequent way.

Now we have two basic layer logic axioms:

A1): statements have truth values only in combination with a layer t = 0,1,2,3,…

A2) Layers are hierarchically ordered, i.e. truth values can be defined using truth values of lower layers,

conversely in lower layers nothing is known about higher (or same) layers.

As a consequence: W(W(A,t)=w,t)=u; W(W(A,t)=w,d)=u für d

Hello William,

I did not check this forum for some month.

Therefore here first my actual thoughts about layer logic,

an answer to your postings will follow later.

Trying to use layers of layers was maybe to ambitios.

Therefore I come back to a easier way:

I will alter the following rule:

A7: layers are upstairs and for themselfes "blind":

W ( W(A,t)=v, d ) = u for t >=d ans any v = u or w or -w.

Now the truth value of "W(A,t)=v" is independant of layers (like d).

As now we see W(A,t) as a (fixed) truth value,

therefore statements about W(A,t) (like „W(A,t)=v“) are statements about truth values an not dependant of layers.

In classic logic a statement A could be substituted by its truth value W(A),

in layer logic A this is possible for every layer t= 0,2,3, ...:

For every layer t statement A can be substituted by W(A,t).

So statements about W(A,t) are de facto classical statements

(where I use a 3rd truth value u "undefined" for symmetrical reasons).

Same with statements about all statements, all layers or about the existence of special properties.

The equality of layer statements is a meta property and easier to define:

W(A=B) = w : for all t: W ( W(A,t) = W(B,t) ) = w. and W(A=B)= f else.

(if A or B are classic and no layer statements, we define W(A,0) :=u and W(A,t):= W(A) else, same for W(B,t))

Equality of layer sets:

W(M1=M2) = W ( For all t: W(xeM1,t) = W(xeM2,t) )

Exspecially: W(M=M)=w .

The succesor set m+ (for the peano axioms) is now more easy:

W(x e m+, t+1) := W ( W(x e m, t) v W(x=m) )

As a whole, layers are just used in a certain "kernel" of logic, the rest remains nearly as usual.

Looking from the perspective of layer logic it still remains a unsolved question,

why in everyday life we so rarely encounter layer effects...

Yours

Trestone

Hello William Jackson,

I followed your links and read about these theories.

But as I hardly understand my own theory, the layer logic,

I confine myself to develop this theory.

When it is a little more stablished, others may use it in different contexts.

As I know little of physics, I am interessted in hints

but may not be able to follow them thoroughly.

Yours

Trestone

Hello,

trying to use layers of layers was maybe to ambitios.

Therefore I come back to a easier way:

I will alter the following rule:

A7: layers for themselfes and for higher layers "blind":

W ( W(A,t)=v, d ) = u for t >=d ans any v = u or w or -w.

Now the truth value of "W(A,t)=v" is independant of layers (like d).

As now we see W(A,t) as a (fixed) truth value,

therefore statements about W(A,t) (like „W(A,t)=v“) are statements about truth values an not dependant of layers.

In classic logic a statement A could be substituted by its truth value W(A),

in layer logic A this is possible for every layer t= 0,2,3, ...:

For every layer t statement A can be substituted by W(A,t).

So statements about W(A,t) are de facto classical statements

(where I use a 3rd truth value u "undefined" for symmetrical reasons).

Same with statements about all statements, all layers or about the existence of special properties.

The equality of layer statements is a meta property and easier to define:

W(A=B) = w : for all t: W ( W(A,t) = W(B,t) ) = w. and W(A=B)= f else.

(if A or B are classic and no layer statements, we define W(A,0) :=u and W(A,t):= W(A) else, same for W(B,t))

Equality of layer sets:

W(M1=M2) = W ( For all t: W(xeM1,t) = W(xeM2,t) )

Exspecially: W(M=M)=w .

The succesor set m+ (for the peano axioms) is now more easy:

W(x e m+, t+1) := W ( W(x e m, t) v W(x=m) )

As a whole, layers are just used in a certain "kernel" of logic, the rest remains nearly as usual.

Looking from the perspective of layer logic it still remains a unsolved question,

why in everyday life we so rarely encounter layer effects...

Yours

Trestone

Hello,

I still do not really know what the „layers“ are,

but cause and effect seem to give a hint:

For cause and effect are (classically) in a hierarchic order,

i.e. the cause has influence on the effect,

but not the other way round.

Same with the layers in layer logic:

A statement in layer t has a truth value and can contribute

to the definition of a truth value of a statement in layer t+1,

but not vice versa.

So we can assign causes to lower levels (like t) and effects to higher levels (like t+1).

With cause-and-effect chains we can construct (almost) arbitrarily high levels.

If we want to start a cause-and-effect chain,

We can use two specialities of layer logic:

On the one hand there is layer 0, the ultimate zero point,

i.e. every chain in layer logic has a natural starting point there

(and no infinite regress necessary).

On the other hand:

How ever high we are in a layer logic chain (with a statement to layer t),

we can come down easily:

We just use the meta statement „W(A,t)=w“, and this statement belongs to layer 1.

(Regarding my last holiday I call this “the Irish slide”).

I think that this resembles in some parts my intuitive understanding of the mind-body relation,

but this here only as a side note.

So even if there remain doubts about concret cause and effect relations (think of Hume!),

those relations are the best examples for “real” layers that I can give today.

Yours

Trestone

Hello,

Up to now I had defined natural numbers in layer logic and set theory

by the following successor function m+:

To every set m we define a successor m+:

W(x e m+, t+1) := W(x e m, t) v W(x=m,1) ( for t=0 without W(x e m, t) )

The (adjusted) Peano axioms hold for m+.

We can define 0, 0+, 0++ etc. this way.

The so defined “natural numbers” m are not constant over layers:

In small layers t

Hello,

perhaps some examples will help to show how layer logic works

(and hopefully inspire somebody to answer or ask a question):

1) a perception statement

A := „I see a red car“.

This classical statement can be true or false: W(A)=w or W(A)=f.

In layer logic this statement has a truth value in every layer t:

W(A,t) = w or f or u.

In practice we will not need different values in different layers for concrete perceptions (and we will not need the third value u).

Therefore we have the spezial case in layer 0 as always: W(A,0)=u

and W(A,t)=w or W(A,t)=f for all t>0.

This is an explanation why we do not have to agree to a layer at concrete statements when determing a truth value.

2) Implicit undefined or self-referencing statement

B:= „This statement is not true“

In classic logic this statement is neither true nor false.

In layer logic we have to modify it slighty, as true is only valid with a layer:

SB:= „This Statement ist true in layer t+1, if it is not true in layer t (and false else)“

It is W(SB,0)=u. Therefore W(SB,1) = W(W(SB,0) -= w,1) = w

Therefore W(SB,2)= W(W(SB,1) -= w,1) = f , W(SB,3)= w, W(SB,4)= w etc.

Here we have a dependency on layers.

In practice the second kind of statements seems to be rather rare,

therefore layers seemed to be unnecessary,

and problems or paradoxes appeared only at the borders of the system.

Wether a logic with layers and less problems or paradoxes is a worthwile proposition

(at least for specialists) may be a matter of taste …

Yours

Trestone

Hello,

neural networks may be a good example,

but what are `layers of abstraction`?

And are there finite numbers of layers?

In layer logic whe have (at least potentially) infinite layers.

Yours

Trestone

Hello,

here some more details about the new set theory,

that can be defined using layer logic:

This "layer set theory" is different in many points to ZFC:

It has only one kind of infinity and the set of all sets is an ordinary set.

The central idea is to treat “x is element of set M” (x e M) as a layer statement:

It is true in layer t+1 that set x is element of the set M, if the statement A(x) is true in layer t.

(There may be still some gaps in my formalization of layer logic and set theory,

but I hope, that this is owing to my limited capabilties and not to gaps in layer logic:

help welcome!)

Equality of layer sets:

W (M1=M2, d+1) = W ( For all t: W(xeM1,t) = W(xeM2,t) , 1 )

Especially: W (M=M, d+1)=w for d>=0.

The empty set 0:

W(x e 0, t+1) := W( W( x e 0, t ) = w , 1 ) = -w for t>=0.

The full set All:

W(x e All, t+1) := W( W( x e All, t ) = w v W( x e All, t ) = u v W( x e All, t ) = -w , 1 ) = w for t>0 and =u for t=0.

So other than in most set theories in layer theory the full set is a normal set.

Axiom M1 (assignment of statements to sets):

W(x e M, t+1) := W ( W ( A(x), t ) =w1 v W ( A(x), t ) =w2 v W ( A(x), t ) =w3 , 1 )

with w1,w2,w3 = w,u,-w

For every layer set M there exists a layer logic statement A(x) witch fulfils for all t=0,1,2, …:

W(x e M, t+1) = W ( W ( A(x), t ) = w v W ( A(x), t ) = -w , 1 )

W(x e M, 0+1) = W ( W ( A(x), 0) = w v W ( A(x), 0 ) = -w , 1 )

= W (u=w v u=-w, 1 ) = -w

Axiom M2 (sets defined by statements):

For every layer logic statement A(x) about a layer set x there exits a layer set M so that for all t=0,1,2,3,… holds:

W(x e M, t+1) := W ( A(x), t ) (or the expressions of axiom M1).

Definition M3 (definition of meta sets):

If F is a logical function (like identity, negation or f.e. FoW(xeM1,t) = W(xeM1,t)=w )

then the following equation defines a meta set M: (M1=M is allowed):

W(x e M, t+1) := W ( F o W(x e M1, t), 1 )

Consequences of the axioms and definitions:

In layer 0 all sets are u:

W( x e M, 0 ) = u (as all statements in layer 0).

In layers > 0:

W(x e M, t+1) := w if W ( A(x), t ) = w else W(x e M, t+1) := -w

For all x and (normal layer) sets M holds: W(x e M, 1) = u (as W(A(x),0)=u).

For all x and meta sets M holds: W(x e M, 1) = w or –w

Last not least let´s look upon the Russell set:

Classic definition: RC is the set of all sets, that do not have themselves as elements

RC:= set of all sets x, with x –e x

In layer theory: W(x e R, t+1) := W ( W ( x e x, t ) = -w v W ( x e x, t ) = u , 1 )

W(x e R, 0+1) = W ( W ( x e x, 0 ) = -w v W ( x e x, 0 ) = u , 1 ) = W (u=-w v u=u , 1 ) = w

Therefore W(R e R,1) = w

W(R e R,2) = W ( W ( R e R, 1 ) = -w v W ( R e R, 1 ) = u , 1 ) = W (w=-w v -w=u , 1 ) = -w

And so W(R e R,3) = w, W(R e R,4) = -w , …

R is a set with different elements in different layers, but that is no problem in layer set theory.

As All, the set of all sets, is a set in layer theory, it is no surprise,

that the diagonalisation of cantor is a problem no more (I just give the main idea):

Be M a set and P(M) its power set and F: M -> P(M) a bijection between them (in layer d)

Then the set A with W(x e A, t+1) = w := if ( W(x e M,t)=w and W(x e F(x),t)=-w )

A is a subset of M and therefore in P(M).

So it exists x0 e M with A=F(x0).

First case: W(x0 e F(x0),t)=w , then W(x0 e A=F(x0), t+1) = -w (no contradiction, as in another layer)

Second case: W(x0 e F(x0),t)= -w then W(x0 e A=F(x0), t+1) = w (no contradiction, as in another layer)

If we have All as M and identity as Bijektion F we get for the set A:

W(x e A, t+1) = w := if ( W(x e All,t)=w and W(x e x),t)=-w ) = if ( W(x e x),t)=-w )

This is the layer Russell set R (I omitted the ´u´-value for simplification)-

and no problem.

So in layer theory we have just one kind of infinity – and no more Cantor´s paradise …

Yours,

Trestone

Hello,

meanwhile I had a look on Goedel´s proof for his first incompleteness theorem.

A central point is a diagonalization argument, that does not hold in layer logic.

Therefore I think, that Goedels incompleteness theorems are not true in layer logic and layer mathematics -

in contrast to classical and constructive mathematics.

One more reason to have a second look on layer logic ...

Yours

Trestone

Hello,

I am also interested in not technical answers,

like what is difficult to understand about layer logik

or what assoziations are started by layer logic?

Yours

Trestone

Hello,

i am feeling a little like "a voice crying in the wilderness",

perhaps I have to wait until " the time has come"?

Meanwhile I am also interested to know, why my layer logic gets less and less answers: Is it the content, the form or both?

Yours,

Trestone