layer logic - alternative for humans and aliens?

[Trestone]

Hi,

 

 

 

looking for a way around the liar and logic contradictions

 

I changed logic (a little bit?) by introducing a new logical dimension:

 

 

 

Statements are not absolutely true or false anymore

 

but true or false related to a viewing angel

 

or kind of logical layer or meta-level.

 

 

 

With this new dimension, problems become solvable

 

that are unsolvable with classical logic.

 

 

 

Most contradictions are not contradicional anymore,

 

as the truth values belong to different layers.

 

 

 

The good news (in my theory):

 

The liar´s paradox, Cantor´s diagonal argument, Russell´s set and Goedel´s incompleteness theorem

 

are valid no more.

 

 

 

The bad news: There is no more absolute truth

 

and we have to get used to a new mathematics

 

where numbers might have multiple prime factorisations.

 

 

 

Over all, infinity and paradoxes will be much easier to handle in layer theory,

 

finite sets and natural numbers more complicated, but possible

 

(but it will be a new kind of natural numbers...).

 

 

 

The theory was in the beginning just a ´Gedankenexperiment´,

 

and my formal description and axioms may still be incorrect an incomplete.

 

Perhaps someone will help me?

 

 

 

Here my axioms of layer logic:

 

 

 

Axiom 0: There is a inductive set T of layers: t=0,1,2,3,…

 

(We can think of the classical natural numbers, but we need no multiplication)

 

 

 

Axiom 1: Statements A are entities independent of layers, but get a truth value only in connection with a layer t,

 

referred to as W(A,t).

 

 

 

Axiom 2: All statements are undefined (=u) in layer 0.

 

VA: W(A,0)=u

 

(We need u to have a symmetric start.)

 

 

 

Axiom 3: All statements in positive layers have either the truth value ´w´ (true)

 

or ´-w´ (false).

 

Vt>0:VA: W(A,t)= either w or –w.

 

(We could have u in all layers, but things would be more complicated).

 

 

 

Axiom 4: Two statements A an B are equal in layer logic,

 

if they have the same truth values in all layers t=0,1,2,3,...

 

VA:VB: ( A=B := Vt: W(A,t)=W(B,t) )

 

 

 

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.

 

 

 

Axiom 7: A statement A can be defined by defining a truth value for every layer t.

 

This may also be done recursively in defining W(A,t+1) with W(A,t).

 

It is also possible to use already defined values W(B,d) and values of meta statements (if t>=1).

 

For example: W(H,t+1) := W( W(H,t)=-w v W(H,t)=w,1)

 

 

 

A0-A7 are meta statements, i.e. W(An,1)=w.

 

 

 

Although inspired by Russell´s theory of types, layer theory is different.

 

 

 

For example there are more valid statements (and sets) than in classical logic

 

and set theory (or ZFC), not less.

 

And (as we will see in layer set theory) we will have the set of all sets as a valid set.

 

 

 

Last not least a look onto the liar in layer theory:

 

 

 

Classic: LC:= This statement LC is not true (LC is paradox)

 

 

 

Layer logic: We look at: ´The truth value of statement L in layer t is not true´

 

And define L by (1): Vt: W(L, t+1) := W ( W(L,t) -= w , 1 )

 

 

 

Axiom 2 gives us: W(L,0)=u

 

(1) with t=0 gives us: W(L,1) = W ( u-=w , 1 ) = -w

 

(2) with t=1 : W(L,2) = W ( -w-=w , 1 ) = w

 

(3) with t=2 : W(L,3) = W ( -w-=w , 1 ) = -w

 

 

 

L is a statement with different truth values in different layers,

 

but L is not paradox.

 

 

 

That should be enough for a start

 

 

 

Yours

 

Trestone

[Trestone]

...

The theory was in the beginning just a ´Gedankenexperiment´,

 

and my formal description and axioms may still be incorrect an incomplete.

 

Perhaps someone will help me?

...

 

Axiom 1: Statements A are entities independent of layers, but get a truth value only in connection with a layer t,

 

referred to as W(A,t).

...

 

 

Now let´s have a look at the 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

[TonyMcC]

It seems to me that you are describing fuzzy logic. If so it isn't really that new.

[Trestone]

It seems to me that you are describing fuzzy logic. If so it isn't really that new.

 

What is it, that makes you think that layer logic is similar to fuzzy logic (see fuzzy logic at wikipedia )?

Layer logic uses (in levels above 0) only true and false as truth values,

while fuzzy logic uses ranges of truth values.

 

My theory must not necessarily be new (as I do not know all about logic), being inconsistent would be worse.

 

Yours,

Trestone

[TonyMcC]

What is it, that makes you think that layer logic is similar to fuzzy logic (see fuzzy logic at wikipedia )?

Layer logic uses (in levels above 0) only true and false as truth values,

while fuzzy logic uses ranges of truth values.

 

My theory must not necessarily be new (as I do not know all about logic), being inconsistent would be worse.

 

Yours,

Trestone

I may be wrong, but you seem to be trying to produce answers which are not necessarily absolutely true or false, but have a likelihood of being true or false. This seems to be an attempt to produce results similar to those produced by fuzzy logic. Layers with two states only have been described in the past. "In fuzzy set theory, classical bivalent sets are usually called crisp sets." to quote from your link.

Edited September 18, 2011 by TonyMcC

[Essay]

My theory must not necessarily be new (as I do not know all about logic), being inconsistent would be worse.

 

Yours,

Trestone

Nor do I know if this is new, and....

 

I'm passing over the symbolic structure you've elaborated since I never studied logic, but to get to the gist....

 

Aren't you just theoretically adding more dimensions or perspectives by which to judge a problem or find a solution (truth)?

 

Sure, when viewed from enough perspectives, you can always find some truth that is valid across a contradiction; but is it a practical perspective?

 

Or put another way....

It seems logical (but maybe not practical) that if you add enough dimensions (as variables) to view a set of problems, then you'll always be able to find a "solution in common" across the set of individual solutions to those problems. But if you limit your dimensions to those of significant or relevant influence, then it'll be more difficult or even occasionally impossible to find a solution-in-common across the set.

===

 

It is often helpful to view multiple problems (such as a contradiction) as if they are multiple, simultaneous, non-linear, multi-variate equations; to see if there is a solution in common across the set of solutions to a given list of problems.

 

Isn't that what you are doing, in a theoretical way, above; but without accounting for the need to limit the multi-variate aspect to only significant or relevant variables? Of course I suppose significance and relevance is too undefined and subjective to limit the theoretical...

 

...so go for it!!!

===

 

It might be helpful in tackling "wicked problems," so we can move on to become a Type I civilization.

 

~

[Trestone]

...

 

Aren't you just theoretically adding more dimensions or perspectives by which to judge a problem or find a solution (truth)?

 

Sure, when viewed from enough perspectives, you can always find some truth that is valid across a contradiction; but is it a practical perspective?

...

 

Hello Essay,

 

that´s the essence of my handling of problems in layer logic!

 

In my eyes I just introduced one new dimension, the logical layer,

that can take digital values 0,1,2,3,...

 

In the complex numbers with the imaginary part as new dimension

the unsolvable problem of the square root of -1 becomes solvable.

 

With the new dimension ´logical layer´ problems like Cantor´s diagonalisation, Russell´s set and Goedel´s incompleteness theorem

become solvable.

 

As with the complex numbers we have to pay by loosing being illustrative and leaving everyday experience.

 

But dealing with paradoxes and infinity without leaving everyday experience might be even more difficult.

 

I am still looking for plausible interpretations for the logical layer,

as I found a method, but would like to have a theory ...

 

Yours,

Trestone

[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

 

 

[Trestone]

Hello,

 

 

 

layer theory looks at first glimpse very similar to "Russells theory of types" - but 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 have a second look ...

 

 

 

Yours

 

Trestone

Edited December 26, 2011 by Trestone

[Ben Banana]

Excuse me as I exclaim without any elaboration, but I really like where this is going! Very interesting, Trestone.

 

I was about to throw together a nice response, but my provincial reality took me away... and now I should go to bed.

 

 

Stay tight;

I hope to return tomorrow.

[Trestone]

I ´ll stay,

 

even if tomorrow never comes ...

 

Yours Trestone

[Ben Banana]

Yes. I desperately tried to make a response the day after I posted that. However, I still need to do my "homework" before replying. By that, I mean how I'm still inable to read the statements you presented. Could you please share a few links or recommend any books that might help? I don't know what's good.

 

Just, please, don't hand over Let Me Google That For You.

 

You mentioned Bertand Russell, Georg Cantor and Kurt Godel. I've heard (and read a bit) about all of them. However, I've never actually read any of their books or published papers. Questions: Have you? Which ones in particular? Would you recommend I read them rather than a dry introduction (such as this)?

 

I guess I'm being a puppy.

 

Would you rather I just find a quick guide and reply as soon as possible? I'm sorry for the let down, but this looks like a good topic and I don't want to make a cheap response.

[Trestone]

Hallo Ben,

 

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

[Trestone]

Hello,

 

my spare time is limited for the next month.

 

One idea I would like to follow are negative layers.

 

(Could be part of a theory for time ...)

 

 

 

 

As in layer 0 all statements are "undefined",

 

the truth value in layer 0 is not dependend from the value in layer -1.

 

Therefore I think the negative branch is completely independend from the positive.

 

All still very speculative.

 

 

 

But I think we have to get clear the basics of the theory first.

 

 

 

Yours

 

Trestone

 

 

[Trestone]

Hello,

 

as the apocalypse seems not to come today, may be I can contribute a small step to the decline of classical logic instead:

 

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<=t

 

We therefore need the value “u” (undefined/unknown) not only in layer 0 but in all layers.

(Vice versa W(A,0)=u now is no longer a isolated formula but a kind of spezial case of W(W(A,t)=w,d)=u für d<=t für d=0)

 

Annother consequence: The equity of layer statements becomes difficult to be proofed:

1st attempt: W(A=B,d+1) := W ( For all t: W(A,t)=W(B,t) , d+1)

after A2: W( For all t: W(A,t)=W(B,t) , d) = W ( For all t<=d: W(A,t)=W(B,t) , d+1) and W(For all t>d: W(A,t)=W(B,t) , d+1) =

W ( For all t<=d: W(A,t)=W(B,t) , d+1) and u.

Therefore W (A=B, d+1) = u if A=B and W(A=B,d+1)=-w if W(A,t0)-=W(B,t0) for t0<d+1.

Equality could so be (sometimes) disproved and never be proofed positively.

 

 

2nd attempt:

Layer statements have to be defined finitely, as they could not be used otherwise.

Often they are recursively defined for layer t+1 using values of statements in layer t.

 

So we can restrain the statements with a finite periodic value pattern over the layers.

Now positive statements over all layers are possible, if the period k (and the advance v) is fully known.

 

Two statements A and B are equal (in layer k+v+1), if they have the same advance and the same period in the layers.

The value for the of equality of two statements is constant for layers t >= v+k+1.

 

Axiom A2) shows that from layer t+1 we only have a perspective to look „downwards“ to layer t or smaller,

the same layer or higher is beyond the information horizon and undefined.

 

The definition of arithmetics now also becomes more complicated, for example the succesor function is usually defined using equality of sets.

 

But why do things easy if they can be done complicated?

 

 

Yours

Trestone

[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 (even if only implicit used) rule:

A7: layers are for themselfes "blind" 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

[ydoaPs]

It looks like an odd way to try to do modal logic. Have you tried proving consitency and completeness?

[ydoaPs]

You mentioned Bertand Russell, Georg Cantor and Kurt Godel. I've heard (and read a bit) about all of them. However, I've never actually read any of their books or published papers. Questions: Have you? Which ones in particular? Would you recommend I read them rather than a dry introduction (such as this)?

You'd be better off reading modern introductions. They are outdated and more difficult than necessary. For example, hardly anyone (even relatively experienced grad students) really "gets" Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I the first time through. I suspect it's vacuous name dropping.

[Trestone]

It looks like an odd way to try to do modal logic. Have you tried proving consitency and completeness?

 

No, I have not tried to proof the consistency or completeness of layer logic up to now.

Formal operations are not my strong side and I am not a mathematican any more ...

 

But as layer logic is very similar to classic propositional logic (just with added layers),

I think that it is both.

 

What would be more spectacular is, that we can define natural numbers with layer logic

and define arithmetics with them (see above).

These "natural" numbers might be a bit different to our common numbers:

I am not sure, if the prime factorization of (very?) large numbers is the same in all layers.

 

And as Cantor´s diagonalization leads (because of the layers) to no contradictions any more,

I hope that the system of layer logic with arithmetics is still consistent and complete,

that is that all true sentences have proofs.

 

But I am still waiting for someone to check this soundly...

 

Yours

Trestone

[ydoaPs]

Without a consistency and completeness proofs, why should we care about any result you get? If the system is inconsistent, you can prove anything and everything.

[Trestone]

Hello ydoaPs,

 

As far as I know it is (according to Gödel) impossible to proof the consistency and completeness of ZFC (in ZFC), the set theory mostly in use today.

 

My set theory based on layer logic might not have this limitations.

 

Of course it would be nice to prove that layer logic is consistent and complete,

but asking for assistance there is one of the reasons why I started the discussion here.

 

Yours

Tresrone

[ydoaPs]

Hello ydoaPs,

 

As far as I know it is (according to Gödel) impossible to proof the consistency and completeness of ZFC (in ZFC), the set theory mostly in use today.

I don't know of any result of Gödel's that would give that implication. Certainly nothing in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.

[Trestone]

Hello ydoaPs,

 

maybe this link will help to understand what I mean (I read just the introduction):

http://www.math.tu-dresden.de/~mbehri/documents/Incompleteness_ZFC.pdf

 

Yours

Trestone

[ydoaPs]

Hello ydoaPs,

 

maybe this link will help to understand what I mean (I read just the introduction):

http://www.math.tu-dresden.de/~mbehri/documents/Incompleteness_ZFC.pdf

 

Yours

Trestone

Yeah, any system strong enough to do both addition and multiplication of natural numbers (which we can do with any reasonable form of set theory) will be incomplete xor inconsistent. You can't prove which one in that system, but that doesn't mean you can't prove which one.

[Trestone]

Yeah, any system strong enough to do both addition and multiplication of natural numbers (which we can do with any reasonable form of set theory) will be incomplete xor inconsistent. You can't prove which one in that system, but that doesn't mean you can't prove which one.

 

Hello ydoaPs,

 

all this argumention and the proofs are based on classical logic.

 

As I showed, Cantor´s diagonalization does not work in "layer logic".

I assume, that Gödel´s proof does not work with "layer logic" and "layer logic arithmetic",

but this I have not proofed up to now (beeing no professional mathematican).

 

Changing the rules of logic is of course a wild thing and a kind of "nasty trick",

("not playing fair"),

but why not give it a try?

 

Yours

Trestone