Article Oldest Paradoxes, Future Mathematics and Mysticism

Dear Professor Blau,

I was very happy when I found this article of you in ResearchGate!

First it showed that you were still activ in philosophic and logic research and second it might open a chance for me to contact you.

When I started to develop an alternative logic about the year 2000, I was driven by similar points as you (me some 25 years later): The liar, Cantor´s diagonalization, Gödel´s incompleteness theorems, Munchhausen Trilemma, mind and body interaction, consciousness.

I had ended my studies at university in mathematics, informatics and philosophy (in Germany: in Würzburg and Marburg) and was a little angry with my math teachers, that they had leaved out Gödel, about whom I learned in philosophy. In philosophy I did not like the high intensity logic was given, so I started a (pseudo) dissertation „Why do we believe in logic?“ (but have not finilazed it up to now).

As I wanted to give an alternative and not only critizise logic, I tried different approaches.

It was a surprise to me, that adding of new parameters (layers , german „Stufen“) and using three instead of two truth values did work rather well.

When I discussed my theory in math and philosophy forums in the internet, I soon learned that it is not easy to get understood with new ideas, and most times I felt like „a voice in the wilderness“.

Therefore I was very happy, when a student told me about you and your reflexion logic. I lost priority, but when I had read your book „Die Logik der Unbestimmtheiten und Paradoxien“ I saw, that your research was in another level. Me more playing around and not always very thoroughly, you state of the art and yet with intuition.

That helped me not to give up and keep searching.

Understanding remains a problem, for I did not understand at least half of your book.

In spite of all this, there is a difference between my „layer logic“ and your „reflexion logic“: I want to replace classic logic by layer logic

(a little classic logic is left in its meta logic),

so all statements have truth values only in connection with layers.

This allows me to define a new set theory, natural numbers and an arithmetics. Here I can show, that the set of all sets is a (ordinary) set and that the diagonalization of Cantor does not longer work. Therefore one kind of infinity, that of the natural numbers, is enough.

As the prime factorizations can depend from the layer, the proofs of Gödel are probably not valid any more – but being not an active mathematican I have not proved this yet.

It is clear to me, that this „knocking on Cantors paradise“

will not be welcomed by all.

And it is rather probable, that my ideas are inconsistent. But I myself can hardly find the fault,

even as my real live profession is test manager. On the other side: „Even a blind hen somtimes finds a grain of corn“.

Here my main ideas to set theory and Cantor´s proof: I define sets (almost) as in naive set theory:

All x that have a property A(x) are in a set M. But what is new and protects from contradictions are the layers:

x is element in set M in layer k+1 if x has property A(x) in layer k.

Vk >=0: Vx VM: W(x e M,k+1):= W(A(x),k)

( W(A,k) is the truth value of A in layer k ).

With this „multilayered“ sets we get the set of all sets as a set

and Russell´s set as a set: If we choose the truth value w (true) for A(x) we get ALL, the set of all sets:

Vk >=0: Vx: W(x e ALL,k+1):= W(w,k)

As W(w,k)=w for all k>0,

x e ALL is true in k+1 for all k>0 and all x.

As (W(w,0)=u (undefined) (as any proposal in layer 0), every x is a „undefined“ element of ALL in layer 1.

As the power set of ALL is ALL, we can use identity to see, that in layer set theory sets and their power set can have the same cardinality.

The Russell set: We choose A(x) as „x is not element of x“

(x e x is false or undefined)

W(x e R, k+1) := ( W(x e x, k)=f or W(x e x, k)=u )

With x=R: W(R e R, k+1) = ( W(R e R, k)=f or W(R e R,k)=u )

Starting with k=0: W(R e R, 0+1) = ( W(R e R, 0) = f or W(R e R,0) = u )

that is W( R e R,1) = w.

k=1: W(R e R, 1+1) = W(R e R, 1) = f or W(R e R, 1) = u,

that is W( R e R,2) = f.

And so on with k it will alternate without contradiction.

If we try the classical proof of Cantor with our set ALL, the set Af of all x, that lie not in f(x) is the layer Russell set. And for other sets the construction of Af will not produce contradictions, as the layers stop them.

All not very elegant but I hope interesting. More details can be find by this link: https://www.researchgate.net/post/Is_this_a_new_valid_logic_And_what_does_layer_logic_mean

And even more (and newer) in German: https://www.ask1.org/threads/stufenlogik-trestone-reloaded-vortrag-apc.17951/#post-492741

Or you might search in the net wit „layer logic Trestone“ or „Stufenlogik Trestone“. Wether you will answer (positive or negative or not - maybe I am five years late?) ,

I thank you for having published reflexion logic and so encouraged me!

Here a short German poem about layer logic (and a English translation) :

Logerick

Es pendelte ein Philosoph mit der Bahn

nahm sich dabei der Logiklücken an

verhedderte sich in Stufen

denn die Geister die er gerufen

verlachen Logik als Wahn.

Logeric: A philosopher commuted by train

examining logic in vain

entangeld within layers

he started spirits and prayers

as many arrived insane.

Yours, Trestone