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.
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 ?)