Re: Flat sets.

On Mar 8, 9:08 pm, Zaljo...@xxxxxxxxx wrote:
      Flat Set Theory:

  Language: First order logic with identity

  Primitives: =,in,[],card

  '[]' to symbolize ordered pair, is a primitive two place function
symbol.

  'card' is a primitive one place function symbol, and it symblizes
'cardinality'.

  Syntax: LaTeX skimmed notation.

  Definition:
  x is atomic  iff  (for all y (y in x implies y=x) and exists y (y in
x))

 1. Axiom of Extensionality: for all z (z in x  iff  z in y) implies
x=y

  Define:
  x is singleton  iff  exists y (y in x  and
  not exists z (z in x  and z neq y))

  2.Axiom of Singletons: x is singleton iff x is atomic

  3.Axiom of Flatness:  y in x implies y is atomic

  4.Axiom schema of Comprehension: If P is a formula in which
  x is not free, then all closures of

  exists x, for all y (y in x iff (y is atomic and P(y)))

  are axioms.

  5.Axiom of Ordered pairs: [x,y]=[z,u] iff (x=z and y=u)

  Define: z #1 [x,y] iff z=x
  Define: z #2 [x,y] iff z=y

  were # is read as 'dimension'

  so 'x #1 [x,y]' is read as 'x is the first dimension of [x,y]'

  6.Axiom of atomicity of ordered pairs: [x,y] is atomic

  7.Axiom of Infinity: [x,y] implies (x neq [x,y]  and  y neq [x,y])

  8.Axiom of atomicity of Cardinals: card(x) is atomic

  Definitions: x equinumerous y  iff  exists f (f:x to y, f is
bijective)
               x subnumerous_to y iff for all f (f:x to y, f is
strictly  injective)
               x supernumerous_to y iff y subnumerous_to x

  9.Axiom of cardinal equality:

    card(x) = card(y) iff x equinumerous y

  Definitions: card(x) < card(y) iff x subnumerous_to y

               card(x) > card(y) iff card(y) < card(x)

  / Theory definition finished.

I do think that this theory is mostly inconsistent, because ordered
pairs here can be of non singleton sets.

However I do believe that if there is a method to RESTRICT the
primitive 2-place function symbole [ ] (of ordered pairs) to be
applicable only for atoms,

We can actually do without that by simply modifying 6 to the
following:

Axiom of restricted atomicity of ordered pairs:

[x,y] is atomic <-> ( x is atomic and y is atomic )

In this way we will avoid the kind of circularity the original theory
with the unrestricted atomicity of ordered pairs would
lead to.

and add
the axiom of atom existence: exists y y is atomic

Yes this axiom should be added, it is axiom.

10) Axiom of atomic existence:

exists y : y is atomic


then I think we can come with a consistent theory.

  Zuhair

.

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