Splitting up logic and sets in levels: a (meta) way around Russell, Cantor and Gödel?

Trestone · science forum beginner Joined: 24 Apr 2006 Posts: 1

Hello,

I have been fascinated by paradoxes for a long time,
especially the liars paradox: "this statement is not true".

In my math study, I was surprised, how complicated set theory has
become since Cantor´s time and how many axioms are needed (p.e. at
ZFC).

In my study of philosophy I heard about Gödel and his proof, that the
very fundaments of mathematics are weaker than I thought.

There were roumors about constructive mathematics, but this seemed to
be an exotic branch - and most mathematicans (including me) seemed
unwilling to restrict themselves to constructivistic methods.

While experimenting with the liars paradox, I looked for a way to
integrate such paradox statements into logic. What was to change?
Soon I found, that using one more dimension could help:
As a statement should not be true and false simultanously, I could
assume that there are actually two different statements: one that it is
true when a second parameter is 1 and annother that is false, when this
second parameter is 2.
I called this parameter "level", usually I symbolize this level
parameter with "t". (But it is not a time parameter, it is more like a
count of meta levels.)

These levels t can be all natural numbers, starting with 0.
(I do not need arithmetics for the levels t.
Perhaps "layer" would be a better word than "level"? In German I call
them "Stufe".)

My proposal for a new logic and set theory with levels turned out to
have interesting features:
It starts with level 0, where all statements are true and not true
simultanously!
Digression: a connection to uncertainty in quantum theory?
In higher levels we have only either true or not true statements
similar to "classical" logic.
But the important point is: statements can have different truth values
in different levels!

Now some more explanations and later we see the more technical part
with the axioms, all still "under construction".
We start with set theory, as I did myself some time ago and look on the
(new?) logic later.

1) Meta Set Theory : sets in levels

We look at Russell´s set R ( R := set of all sets, that are not
elements of themselves ).
"x e R <-> x -e x" and the paradox for x=R: R e R <-> R -e R !

My idea is to use "meta set theory" and add an level parameter to the
element operator "e" at the left and right site:
" x e(t+1) R <-> x -e(t) x ". x is element of R in level t+1 iff x is
not element of R in level t.

As all sets are empty at level 0 (see below) we can now look what
happens, if we use R for x:
R -e(0) R is true -> R e(1) R is true -> R e(2) R is not true -> R e(3)
R is true -> R e(4) is not true -> ...
We now better understand the Russell antinomy: In classical set
theories we have only one dimension and can not express a oscillating
behavior, in meta set theory there is no problem, thanks to the levels.

Of course my set R is not exactly the set of Russell, I even use a
different set theory:
In meta set theory sets are dimensionless "things-in-itself" which have
properties (like being element or being equal) that we can observe only
by levels - and the properties can be different in different levels.

A second point is essential: not all properties of a set in level t can
be described/decided in level t, in most times we need a "higher"
(meta-) level.
The most important property "x e(t) M" we can not decide in level t but
in level t+1.
It is as if in level t we can "see" all properties of lower levels, but
not of level t itself or of higher levels, we are "blind to ourself and
upward".
This has interesting consequences: The maximum of a set in level t is
not a property in level t but in t+1.
The levels are not independent, usually there are relations between
properties in level t and level t+1: Otherwise we would have to deal
with too much infinty...

Now the axioms:

1.0 Notation:

V means "for all"
E means "it exists at least one"
<-> means "if and only if"
e(t) means "is at level t element of"
-e(t) means "is at level t not element of"
=(t) means "is at level t equal to"
= means "is equal to " (for all levels)

1.1a axiom of the level-set T (metaset T of levels):

There exists a set of levels T which is inductive, i.e. 0 is a element
of T and with every element t its succesor t+1 is element of T.

(T is like N with Addition, at T we need only one (or no) level. T is a
axiomatic metaset and perhaps a weak point of meta set theory.)

1.1b axiom of the level upward blindness

In level t we can "see" all properties of lower levels, but not of
level t itself or of higher levels. The truth of a statement about a
property in level t is not known in level t or lower, but in level t+1
and higher.

Usually there is a connection (usually a formula as in axiom 1.4)
between a property in level t+1 and level t, for example the liar´s
paradox is true in level t+1 if it is not true in level t.
For using induction on t we only need a starting point. As anything is
true in level 0 we get this start for free. These fixed connections are
needed to avoid infinity. but i did not formulate an axiom yet.

In the following text t is always a level, i.e. a (meta-)element of T.
x,y,z,a,b are sets , A(t,y), F(t,y) are functions or terms with a fixed
level t.
(All sets are sets of the meta set theory with levels, other sets are
called "classic sets")

1.2 axiom of extensionality

Vt: Vx: [(x e(t) y) <-> (x e(t) z)] <-> y =(t) z
set y and z are equal in level t iff z and y have the same elements in
level t.
y=z <-> Vt: y =(t) z
set y and z are equal iff z and y are equal in all levels.
(Usually we show this by induction on t)

1.3 axiom of level 0

Vx: Vy: x -e(0) y
At level 0 all sets are empty.
(Perhaps they are full simultanously: Vx: Vy: x e(0) y ?)

1.4 axiom of existence

Vt: Ey: Vx: ( x e(t+1) y <-> A(t,x) )
Other form: Vt: y =(t+1) {x: A(t,x)}

If A(t,x) is a "set-term" of level t ( for example (x e(t) x) or (x
=(t) x) )
there exists a set y that fulfills for all levels this set-term (in
t+1).

Definition of set-term:

1.4.1 ,,0" is a set-term of any level (empty set, constant)
1.4.2 ,,x" is a set-term of any level (identity)
1.4.3 If ,,a" and ,,b" are set-terms of level t, then same for
,,-a", ,,avb", ,,a^b",
,,a->b", ,,a<->b", ,,Ex: a(x)", ,,Vx: a(x)"
1.4.4 If v,w are sets then ,,v e(t) w" and ,,v =(t) w" are
set-terms of level t
1.4.5 BEWARE: If a,b are set-terms of level t, then same for
,,Vd<t+1: a(d)(x)" and ,,Ed<t+1: a(d)(x)" , but mostly NOT for
,,Vt: a(t)(x)" oder ,,Et: a(t)(x)"
(Properties belong to levels, there is no level for all levels ...)

The terms that can be constructed using 1.4.1 - 1.4.5 are the
"set-terms".

Lemma 1: The empty set "0" and the full set "ALL" exist.
Proof: Use "x -=(t) x" and "x=(t) x" in axiom 1.4 #

Lemma 2: At level 0 all sets are equal, especially ALL =(0) 0.
Proof: Use axiom 1.2, axiom 1.3 and lemma 1 #

Lemma 3: The set R of Russell with Vt:[x e(t+1) R] <-> [ x -e(t) x ]
exists.
Proof: Use "x -e(t) x" as A(t,x) in axiom 1.4.
As x -e(0) x because of axiom 1.3, we have R -e(0) R, so R e(1) R, R
-e(2) R , R e(3) R, ... #

Lemma 4: Let P(M) be the power set to a set M with:
x e(t+1) P(M) :<-> Vy: y e(t) x -> y e(t) M.
Then P(M) exists to every set M and there are sets A which are in any
level t+1 in bijection to their power set P(A).

A set x is element of the power set of M in level t+1 iff x is subset
of M in level t, i.e all elements of x in level t have to be elements
of M in level t.
Always true: 0 e(t+1) P(M) und M e(t+1) P(M).

Proof: existence is proved with axiom 1.4.
Now to Cantor:
Of course the power set in meta set theory is different to the power
set of classic set theory (f.e. ZFC), but it is as similar constructed
as possible.
We look at A = ALL. We want to show: P(ALL) = ALL with identity as
bijection.
At level t+1 we have by definition of P(ALL):
x e(t+1) P(ALL) :<-> Vy: y e(t) x -> y e(t) ALL.
We have to show: x e(t+1) ALL <-> Vy: y e(t) x -> y e(t) ALL.
This is true as every set is element of ALL in all levels. (Level 0 no
problem).#

As this proof is too simple we will look now more in detail where is
the difference to Cantor and his famous proof about the (classic) power
set.
Cantor uses a bijection f: M -> P(M) to construct a paradox (classic)
set:
A(f) := {x e M: x -e f(x)}. As A(f) is a subset of M there has to be a
element m of M with f(m) = A(f). But both m e f(m) and m -e f(m) lead
to a contradiction!

Now the same construction in meta set theory:
We assume a bijection f in level t+1: f: M ->(t+1) P(M).
We construct A(f) as: x e(t+1) A(f) :<-> x -e(t) f(x) und x e(t) M .
A(f) is a set, its elements in level t+1 are elements of M in level t.
BEWARE: we could not define A(f) with this definition in level t
because of axiom 1.1b.
Now let us assume, there is m e(t) M with f(m) =(t+1) A(f).
If m e(t+1) f(m) then m e(t+1) A(f) -> m -e(t) f(m) , no contradiction
as annother level.
If m -e(t+1) f(m) then m -e(t+1) A(f) -> m e(t) f(m) , no contradiction
as annother level.
In our example with M=ALL and f=identity we find A(f) = R, the set of
Russell:
x e(t+1) A(f) <-> x -e(t) f(x) <-> x -e(t) x <-> x e(t+1) R.

1.5 Natural numbers and arithmetic

A set theory without natural numbers and arithmetic is not very
interesting, but I was not able so far to close this gap.

As a start I defined:

Lemma 5: For every set m we have a succesor m' defined as
x e(t+1) m' :<->( x e(t) m ) v ( x = m )

Of course I hope that natural numbers and arithmetic are possible (with
not too many new axioms, best none). And it would be nice, if we could
get around Gödel the same way as around Russell and Cantor...

Part 2 with meta logic will follow soon.

Gruß
Trestone.

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