🚀 go-pugleaf

On Feb 7, 12:28 am, Bill Taylor <wfc.tay...@gmail.com> wrote:
> 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.
>
> So my query is:
>
> 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!

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