Graham Cooper
2012-02-16 23:00:00 UTC
I'm not sure that this is a very unambiguous query,
so please be gentle anfd as helpful as possible.
The motivation for my query is the apparent circularity in trying
to found set theory on a model theory that is itself a set theory,
though metamathematical and much smaller than ZFC itself or even Z.
What is the *minimal* set theory required to formalize the ideas
of models (of FOL theories), satisfaction, soundness, and so forth?
Presumably it doesn't require power set, but needs at least unions
and finitely-adic cross products. How much of separation is needed?
TIA, -- baffled Bill
** Set theory is a shotgun marriage between powerset & well ordering.
You don't need a CONSTRUCTABLE SET UNIVERSE at all!so please be gentle anfd as helpful as possible.
The motivation for my query is the apparent circularity in trying
to found set theory on a model theory that is itself a set theory,
though metamathematical and much smaller than ZFC itself or even Z.
What is the *minimal* set theory required to formalize the ideas
of models (of FOL theories), satisfaction, soundness, and so forth?
Presumably it doesn't require power set, but needs at least unions
and finitely-adic cross products. How much of separation is needed?
TIA, -- baffled Bill
** Set theory is a shotgun marriage between powerset & well ordering.
MATHEMATICS ITSELF IS NOT SOME PHANTOM PYRAMID of ladders of ordinals.
You can avoid Russell's Paradox in a modified NAIVE SET THEORY where
LOGICAL_CONSTRUCTIONS must be PROVEN or at least PROVABLE!
ZFC does NOT HAVE AN AXIOM_OF_SPECIFICATION!
Separation is an AXIOM_OF_SUBSET_SPECIFICATION
The axiom of SPECIFICATION is really just SUBSET SPECIFICATION.
You NAME the SUBSET Y according to the definition of it's elements
(ONCE THE SUPERSET Z ALREADY EXISTS)
A(z): A(p1,p2..pn):
E(y): A(x): x e y <-> (x e z & P(x,y,p1,p2..pn))
P is a predicate (which can be recursively defined at a lower level)
in ZFC
***
Before the AXIOM OF SPECIFICATION was NAIVE SET THEORY
According to naive set theory, any definable collection is a set.
E(y):A(x): x e y <-> P(x)
This yields an INCONSISTENT theory since replacing P(x) with x ~e x
yields
z e z <-> z ~e z
***
So SPECIFICATION resolves Russell Set and ZFC is CONSISTENT for now.
But: the paradox of Russells Set is just pushed down one level!
***
Here in DCPROOF the Russell Set is proven to not exist.
Here we prove that the set of all sets that are not elements of
themselves does not exist.
Prove: ~EXIST(s):ALL(x):[x e s <=> ~x e x]
Proof
-----
Suppose to the contrary...
1 EXIST(s):ALL(x):[x e s <=> ~x e x]
Premise
Define: r
2 ALL(x):[x e r <=> ~x e x]
E Spec, 1
Apply the definition of r to itself - CONTRADICTION
3 r e r <=> ~r e r
U Spec, 2
As Required:
4 ~EXIST(s):ALL(x):[x e s <=> ~x e x]
4 Conclusion, 1
*****************************************
OK, remember that?
~EXIST(s):ALL(x):[x e s <=> ~x e x]
There does NOT EXIST a set R of all x that are not elements of x.
****************************************
Here's another proof in ZFC
Example 8: The Paradox of the Universal Set
-------------------------------------------
The so-called Universal Set is the set of all things.
Here we will show that, it cannot exist.
Prove: ~EXIST(s):[Set(s) & ALL(a):a e s]
Prove: ALL(u):~[Set(u) & ALL(a):a e u]
Suppose...
1 Set(u) & ALL(a):a e u
Premise
2 Set(u)
Split, 1
3 ALL(a):a e u
Split, 1
Apply the Subset Axiom for u *AXIOM OF SPECIFICATION*
4 EXIST(s):[Set(s) & ALL(a):[a e s <=> a e u & ~a e a]]
Subset, 2
Define: r, the set of all things that are not elements of themselves.
This results in the SAME CONTRADICTION that disproved Russell Sets
before!
3 r e r <=> ~r e r
So, the NON-EXISTENT SET R is DEFINED USING AXIOM OF SPECIFICATION
as a subset of THE UNIVERSAL SET U
and it "double-proves" that the set U cannot exist either!
****************************************************************
http://tinyurl.com/new-set-theory
A SIMPLE SET THEORY (WITH NO GODEL STATEMENTS)
According to naive set theory, any definable collection is a set.
E(y) A(x) x e y <-> P(x)
This yields an INCONSISTENT theory since replacing
P(x) with x ~e x
results in the contradiction by forming Russell's Set
z e z <-> z ~e z
***
E(y) A(x) x e y <-> P(x) ----- NAIVE SET THEORY
PESSIMIST SET THEORY
[E(y) A(x) x e y <-> P(x,y)] & PRV[E(y) A(x) x e y <-> P(x,y)]
In PESSIMIST Set Theory, Russell's Set and the Axiom Of Infinity and
Godel
Statements cannot stratify as a premise as they are not provable by
PRV().
PRV(c) <-> c v (E(a) E(b) a&b->c & PRV(a) & PRV(b))
PRV() is thought to be unprogrammable in conventional logic, but in
PESSIMIST SET THEORY and OPTIMIST SET THEORY Godel Statements will not
even stratify *they have no proof* so theories that only stratify
given PRV() are 'bona-fide' consistent!
G <-> !PRV(G)
= ! (G v (E(a) a->G & PRV(a)))
-> ! (G v FALSE)
-> !G
CONTRADICTION
NOT(PRV(G))
NO GODEL STATEMENTS!
***
OPTIMIST SET THEORY
[E(y) A(x) x e y <-> P(x,y)] & !PRV[!E(y) A(x) x e y <-> P(x,y)]
In OPTIMIST Set Theory, the Axiom Of Infinity can stratify as a
premise as there is no proof otherwise PRV(~INF).
ZFC AXIOM OF INFINITY
E(y) A(x) 0 e y & (x e y -> S(y) e y)
------------------------
In OPTIMIST SET THEORY (OST)
[E(y) A(x) x e y <-> P(x,y)] & !PRV[!E(y) A(x) x e y <-> P(x,y)]
PREMISE: E(INF) A(x) x e INF <-> P(x, INF)
& ~PRV(~E(INF) A(x) x e INF <-> P(x, INF) )
ASSUME: ~PRV(~INF) *there is no proof INF does not exist
-> E(INF) A(x) x e INF <-> P(x, INF) #1
where
P(x, INF) <-> (x = 0) v (E(z) x = S(z) & P(z, INF))
e.g. P(3, INF)
--> P(2, INF)
--> P(1, INF)
--> P(0, INF)
--> TRUE
3 e INF <-> P(3, INF) x=3 -> #1
3 e INF <-> TRUE
3 e INF
INF is just the infinite set N {1,2,3,...}
Rather than use a special axiom Infinity is just defined using an
induction predicate.
P(x, INF) <-> (x = 0) v (E(z) x = S(z) & P(z, INF))
***
Herc
--
There are 2 types of Naive ~ Optimism and Pessimism
