looking for a way around the liar and other paradoxes
I focused on logic as the culprit:
By introducing a new logical dimension, most problems can be solved.
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.
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
(not yet fully proved) 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.
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.
(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) or ‘u’ (undefined).
Vt>0:VA: W(A,t)= either w or –w or u.
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.
And self referring statements and sets are allowed in layer logic and layer set theory.
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 iff W(L,t) -= w and W(L, t+1:=-w else.
Axiom 2 gives us: W(L,0)=u
(1) with t=0 gives us: W(L,0) = u-=w -> W(L,1) = w
(2) with t=1 : W(L,1) = w -> W (L,2) = -w
(3) with t=2 : W(L,2) = -w-=w -> W (L,3)= w ; W(L,4)= -w; etc.
L is a statement with different truth values in different layers,
but L is not paradox.
That should be enough for a start
Now what do you think:
Is this ´changing the rules´ with the help of a new dimension worth to be tried,
or too wild a shot?