🚀 go-pugleaf

On Feb 18, 4:19 am, "DonH" <donlhumphr...@bigpond.com> wrote:
> "Graham Cooper" <grahamcoop...@gmail.com> wrote in message
>
> news:2e515870-d95b-4dfc-89c6-80a7dbe5d793@vh10g2000pbc.googlegroups.com...
> 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
>
> # When I was a kid, I didn't hear of "sets" at school, there was only the
> definition and classification of things - and a "class" of objects had some
> common attribute(s).
>     What then is a set?  It is a group of objects which may, or may not,
> have things in common - a bit like contents of box of junk in the attic, or
> goods on a jumble sale at the church fete.
>     A set is a mental concept, and has no reality apart from that.
>     You can count the number of objects in a set, but a set can only be a
> member of itself if lumped into all the non-set aspects of the universe.
> This may be seen as highly sophisticated, or road to madness, depending on
> viewpoint.
>     Words are human artifacts which we apes like to monkey around with.
>     Most inhabitants of lunatic asylums live in a (mental)
> world-of-their-own.  Wanna join?


You can always work with Natural numbers and the single Relation "e"
for a more concrete method.

TABLE_e
Element | Set
1          1
2          1
123        345
...


http://a5.sphotos.ak.fbcdn.net/hphotos-ak-ash4/s320x320/375336_2461864784302_1182660270_32343844_806546620_n.jpg


Herc