Trestone

Hello, Imagine, the SETI-Project has reached contact to something about 5 light years in distance and we switch some of the first years, so the communication with the aliens is in English. One of the researchers (“SEARCH”) is logician and mathematician, as those fields are supposed to be of universal validity. Here the protocol of the (nearly) first contact. Yours Trestone: SEARCH: “Hello ALIEN, we are especially interested in your logic and mathematics and wether they are different to ours?” ALIEN: “Hello SEARCH, we do not have one logic or mathematics. We use different ones for different purposes.” SEARCH: “Can you give me an example for such a logic?” ALIEN: “Just give me some problems you want to handle, and we will find a suitable logic for you.” SEARCH: “First all statements should be either true or false and implications can be evaluated by analyzing the components. It should help for consistant argumentation and reasoning.” ALIEN: “Human classical logic would be a good choice, but not all statements would be either true or false. By the way we use this logic in communicating with you.” SEARCH: “With the execeptions, do you think of statements like the liar statement:“This statement is not true”?” ALIEN: “Yes, and with this logic you will have mathematical restrictions like the incompleteness theorems of Kurt Gödel or the set of all sets being no set.” SEARCH: “You know Kurt Gödel?” ALIEN: “We studied all that you have sended to us.” SEARCH: “As we tried to do. Could you show me a logic without the restrictions you mentioned?” ALIEN: “You could do it easily yourself: The logic “everything is true””. SEARCH: “Ok, that is true, but I meant a more useful example for practical purposes?” ALIEN: “We tried a “joke”! A logic of the kind you asked for is not to complicated but a little bit technically boring. You have to use additional dimensions. It is similar to solving the square root of -1 with complex numbers.” SEARCH2: “Just try do explain it to me. By the way I am a new human being, as my collegue died of old age.” ALIEN: “Hello SEARCH2! Perhaps we should give longer answers to you … For analyzing all three problems indirect proof is classically used. So there are statements which would be simultaneously true to their negations. In the new logic these statements (or more precisely their truth values) are in another dimension than the negations. We call this dimensions layers and the logic “layer logic”. There are indefinitly many layers k=0,1,2,3,… and every statement has a truth value in every layer. The truth values can be different in different layers. Classic statements are similar to layer statements that are constantly true (=T) or constantly false (=F) in all layers greater than 0. In layer 0 all layer statements are undefined (=U, a symmetrical starting) and we have “undefined” as a third truth value in all layers. All layer statements need a truth value in every layer and truth values do only exist for the combination of statements and layers. Truth values can be defined recursivly using already defined statements and smaller layers.” SEARCH2: “Let us try an example, the statement “This statement is not true”.” ALIEN: “First we have to add layers, as a statement alone has no truth value: “This statement L is not true in layer k”. Now we have to define a truth value for L in every layer. We do this by defining when L is true for every layer k+1 depending on the truth value of L in layer k: For every k=0,1,2,…: L is true in layer k+1 if L is not true in layer k and L is false else. With v(L,k)=T for “L has truth value true in layer k”: v(L,k+1):=T IF ( v(L,k)=F or v(L,K)=U ) ELSE v(L,k+1):=F We have v(L,0)=U as all statements are undefined in layer 0. v(L,0+1):=T IF ( v(L,0)=F or v(L,0)=U ) ELSE v(L,0+1):=F v(L,0+1):=T IF ( U=F or U=U ), therefore v(L,1)=T v(L,1+1):=T IF ( v(L,1)=F or v(L,1)=U ) ELSE v(L,1+1):=F v(L,1+1):=T IF ( T=F or T=U ) ELSE v(L,1+1):=F, therefore v(L,2)=F So we have v(L,0)=U, v(L,1)=T, v(L,2)=F, v(L,3)=T, v(L,4)=F, … SEARCH2: “What does this mean for the original liar statement, is it true or false?” ALIEN: “Not all layer statements are classical statements, the liar statement is one of those nonclassical statements. It has no classical truth value, but is a normal layer statement with alternating truth values. It is like a complex number that is not real. To get the benefits of layer logic you have to use it. SEARCH2: “But it is not easy for me to change to a new logic, for example if we talk about it we should use a known logic.” ALIEN: “Fortunately we can use human classic logic when talking about layer logic, as this logic is the meta logic of layer logic.” SEARCH2: “Is layer logic similar to the theory of types by Bertrand Russell?” ALIEN: “In the theory of types objects are splitted into differend types and the types are used to avoid self reference within objects. In layer logic the truth values are splitted into different layers and the layers enable us to have self reference within objects and statements. So the answer is mostly no.” SEARCH2: “Can you give an example for sets and self reference?" ALIEN: “So let us have a look on layer set theory, a rather nice piece of work. The central idea is to treat “x is element of set S” (x e S) as a layer statement: It is true in layer k+1 that set x is element of the set S, iff the statement A(x) is true in layer k. v(x e S,k+1) :=T if v(A(x),k) = T (and F or U else). And as in the original theory of Cantor for every set statement A(x) there exists a set. We have the following two rules for sets: Rule M1 (assignment of statements to sets): For all k,sets x,set M exists a set statement A(x) which fulfills: v(x e M, k+1) := v(v( A(x), k)=w1 v v(A(x), k)=w2 v v(A(x), k)=w3,1) with w1,w2,w3 = T,U,F or one or two of them. Rule 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 k=0,1,2,3,… holds: v(x e M, k+1) := v( A(x), k ) (or the expressions of rule M1). You asked for examples: The empty set 0: We use “x e 0” as A(x) For all k>=0: v(x e 0, k+1) := v(v( x e 0, k )=T,1) (=F for k>=0) v(x e 0, 0+1) := v( v( x e 0, 0 ) = T, 1) = v( U = T , 1 ) = F v(x e 0, 1+1) := v( v( x e 0, 1 ) = T, 1) = v( F = T,1) = F, etc. The full set All: v(x e All, k+1) := v( v( x e All, k ) = T v v( x e All, k ) = U v v( x e All, k ) = F , 1 ) = T for k>0 and =U for k=0. v(x e All, 0+1) := v( v(x e All, 0) = T v v(x e All, 0) = U v v v(x e All, 0) = F, 1 ) = = v( U = T v U = U v U = F , 1 ) = T v(x e All,1+1) := v(v( x e All, 1) = T v v(x e All, 1) = U v v v( x e All, 1) = F , 1 ) = = v( v( T = T v T = U v T = F , 1 ) = T, etc. So other than in most set theories in layer theory the full set is a normal set.” SEARCH2: “What is with the Russell set, the set of all sets that are not elements of themselfes?" ALIEN: “We translate the definition of the Russell set R to layer set theory: v(x e R, k+1) := v( v( x e x, k ) = F v v( x e x, k ) = U , 1 ) v(x e R, 0+1) = v( v( x e x, 0 ) = F v v( x e x, 0 ) = U , 1 ) = T (U=F v U=U , 1 ) = T ; therefore v(R e R,1) = T v(R e R,2) = v( v( R e R, 1 ) = F v v( R e R, 1 ) = U , 1 ) = F (T=F v F=U , 1 ) = F; therefore v(ReR,3) = T, v(ReR,4) = F, ... R is a set with different elements in different layers, but that is no problem in layer set theory, so R is a layer set." SEARCH2: “I suppose that Cantor´s diagonalization in layer theory is not valid any more?” ALIEN: “You are right. The set of all sets All is in bijection (via identity) with its power set. So we do not need different kinds of infinity in layer set theory. But let us have a look into the proof of Cantor, transferred to layer theory: Be S a set and P(S) its power set and F: S -> P(S) a bijection between them (in layer d). Then the set A with v(x e A, k+1) = T := if ( v(xeS,k)=T and v(xeF(x),k)=F ) is a subset of S and therefore in P(S). So it exists x0 e S with A=F(x0). First case: v(x0 e F(x0),k)=T , then v(x0 e A=F(x0), k+1) = F (no contradiction, as in another layer) Second case: v(x0 e F(x0),k)= F then v(x0 e A=F(x0),k+1) = T (no contradiction, as in another layer) If we have All as S and identity as Bijektion F we get for the set A: v(x e A, k+1) = T := if ( v(x e All,k)=T and v(x e x),k)=F ) = = if ( v(x e x),k)=F ) This is the layer Russell set R (We omitted the ´u´-value for simplification) - and no problem.” SEARCH2: “And can we still do arithmetics?” ALIEN: “Yes, mostly as usual, sometimes in a special way. Let us start with the Peano axioms: We can define the successor m+ of a set m in the following way: v(x e m+, k+1) := v(x e m, k) v v(x=m,1) For k=0 without v(x e m, 0): v(x e m+, 1) := v(x=m,1) We start with m=0, v(0+,1) = v(x=0,1): In layer 1 the only element of 0+ is 0. v(x e 0+, 1+1) := v(x e 0, 1) v v(x=0,1) = F v v(x=0,1). v(x e 0+,2+1) :=v(x e 0,2) v v(x=0,1)= F v v(x=0,1) = v(x=0,1) So 0+ is a set with only element 0 in all layers >=1. Now we look at m=0+ v(x e 0++, 1) := v(x=0+,1): In layer 1 the only element of 0++ is 0+. v(x e 0++, k+1) := v(x e 0+, k) v v(x=0+,1) In all layers >1 the only elements of 0++ are 0 and 0+. So we find: n+ contains in layer 1 exactly the element n n+ contains in layer 2 exactly the elements n, n-1 n+ contains in layer n exactly the elements n, n-1, …,1 n+ contains in layer k>n exactly the elements n, n-1, …,0 For large k the natural numbers of layer set theory are therefore similar to the classical natural numbers. The (adjusted) Peano axioms hold for m+. We can define 0, 0+, 0++ etc., (the natural numbers) this way. The addition of numbers we define using the successors: v(x e n + m+, k+1) := v(x e (n+m)+, k+1) = = v(x e (n+m),k) v v(x=(n+m),1) Multiplication: v( x e n*m+, k+1 ) := v( x e n*m + n, k+1) = = v(x e (n*m + n-1)+, k+1 ) = = v( x e (n*m + n-1), k) v v(x = (n*m + n-1),1) v(x e 2*2+, k+1 ) =v(x e 2*2+2, k+1 ) =v(x e (2*2+1)+, k+1)= = v( x e 5, k) v v(x=5,1)" SEARCH2: “Can you give me more details in a special paper?” ALIEN: “You already have it: For first fundaments look at a Review of the logic of Prof. Ulrich Blau ( as it is a pdf-file, you may have to put this URL directly in your browser: https://wwwmath.uni-muenster.de/u/rds/blau_review.pdf ) and for layer logic at a thread by Trestone at ResearchGate: https://www.researchgate.net/post/Is_this_a_new_valid_logic_And_what_does_layer_logic_mean Or you may search “the net” with “layer logic “Trestone”“ or with “Stufenlogik Trestone” (in German). The symbolization there is slightly different: W(A,t) is used instaed of v(A,k). There still is no academic paper for layer theory – perhaps someone is interested to do this?” SEARCH2: “It will probably not be me, as my time is fading out …" AlIEN: “Hello SEARCH2, you did not ask a question?” ALIEN: “?” ALIEN: “Here an aspect that might be interesting for philosophers: The Münchhausen foundation trilemma (Agrippa`s trilemma), that there are only three poor choices to fundament and start our argumentations gets a new option with layer logic: If we assume that a reason has to be true in a higher level than the founded, the reasoning can go back not further than to layer 1. As every reasoning reduces the layers at least for 1, starting at an arbitrary layer we reach layer 1 after finite steps.” ALIEN: “?” ALIEN: “Hello, is there anybody out there interested to continue this communication?”

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, 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<m m has less elements than in large (where it becomes constant) and similar to the classical natural numbers. But we can use an alternative definition, that is not so hierarchical: If m is defined in layer t+1, we can use values regarding m and layer t+1 to define m´ in layer t+1: W(x e m+, t+1) := W(x e m, t+1) v W(x=m,1) This definition is nearer to the classical natural numbers and I think we get sets, that are not not layer dependent. I have not checked all Peano axioms yet. We might do this overall and reduce the use of layer hierarchie to critical cases (like self reference and undefinedparts). Whether we get back some of the classical problems (like Gödel´s uncompleteness theorem) by this way I do not see so far … 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, the results of Cantor and Gödel are astonishing. But we still can ask, what we can do to avoid the consequences (and perhaps expulse some mathematicans from Hilbert`s paradise). As physics has showed us with time, space and matter, a small change in basic concepts may alter a lot. In mathematics we have the complex numbers that show how to solve unsolvable equations like x*x= -1. As (classical) logic is used in the proofs of Cantor and Gödel, I looked for alterations in logic, that would influence these proofs and give us other results. As I always was doubtful to indirect proofs (where there are two branches), I experimented with different “views” in logic, where every “view” has its own truth value for a proposition. (For example “the liar” is “true” from view1 and false “from” view2). Later I called the views “layers” and used (inductive) natural numbers t = 0,1,2,3,… Now a proposition or statement is not longer “true” (ex-)or false”, but has a truth value only in combination with a layer t, and the truth values in different layers can be different. The layers have some hierarchical order, but the whole is quite different to the “Hierarchy o Types” by Russel, as self reference is fully allowed. Of course this is not classic logic any more, but it has nice effects to set theory: In Cantor`s diagonalization proof we can see, that different layers are implied and therefore there is no contradiction. We therefore need no different kinds of infinity, the “set of all sets” is without paradoxes (and is its own powerset), even “Russell´s set” is existing witout contradiction. We even need less axioms than ZFC to form a set theory. More Details are in the neighbouring thread: http://www.scienceforums.net/topic/59914-layer-logic-a-new-dimension/ The Peano axioms and an arithmetic can be defined with layer logic (but the uniqness of the prime factorization might not hold), I suppose that Gödels proof for incompletness does not hold for layer logic and layer set theory, but beeing no professional mathematican I did not look deeply up to now. Perhaps someone might do this? (And formalize my more intuitive settings). Yours Trestone

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

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

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

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

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

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

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

Hello, maybe there is some complication: we at most know, that we know nothing (Sokrates, the wisest of all man according to the oracle of delphi) so how can we know that there is somthing like us, the world or logic? As long as a fool can ask more questions than thousand wise ones can answer, there is (logically?) no fear that all answers will be found. But perhaps we should be open to a surprisingly other kind of solution, like playing ping pong and having fun. Yours Trestone (an amateur ping pong player)

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

I ´ll stay, even if tomorrow never comes ... Yours 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

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, I see one more (philosophical) problem: If nothing existed, may be logic would not exist as well, as we learned with time in physics ... For I think that logic has developed with mind and could therefore be dependent from the existence of something. What the (illogical?) consequences for "if nothing existed" are ore would be (if any) , I do not know ... Yours Trestone

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

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

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

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