Flat sets.


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, and add
the axiom of atom existence: exists y y is atomic
then I think we can come with a consistent theory.

Zuhair




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