# [FOM] Simple Para-Type Theory "SPT".

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Sun Jun 26 15:19:25 EDT 2011

```The following theory is defined with the aim of being equi-consistent with
TSTU+infinity. It doesn't use typed language, nor any restriction on
quantifiers, and it is defined in first order logic with equality and
membership, so its defined in the same language of Z. The ontology of this
theory is that of sets and ur-elements, the later are defined as distinct
empty objects. Extensionality of this theory is the same as in NFU, i.e.
any two non empty sets are equal if and only if they have the same
members. The structure of the theory is very simple, we have infinitely
many empty objects, the set of all those objects is the base tier, then
upper tiers of sets are constructed by a special sort of powering starting
from the base tier. Separation holds as in Zermelo's. All sets belong to
some tier.

Axioms:

1.Extensionality: as in NFU

2.Base: Exist x. for all y. y in x iff not Exist z. z in y

Define: x=O iff for all y. y in x iff not Exist z. z in y

3.Power: for all s. Exist x. for all y. y in x iff y is non empty subset of s

Define: P(s)=x iff for all y. y in x iff y is non empty subset of s

4.Separation: as in Z.

5.Axiom of Regularity: for all x. Exist i. x in P_i (O).

where P_i (O) is defined recursively as:

P_i (O) = O, when i=0
P_i (O) = P( P_i-1(O)), for every natural number i>0.

natural numbers are defined in the usual way done in type theories.

Necessary forms of Choice can be added to the above theory.

Since most ordinary mathematics can be implemented in TST(U)+infinity.
Then if the above theory is equi-consistent with TSTU+infinity, then it
can be used to implement ordinary mathematics. The basic merit of this
theory is that it has simpler structure than typed theories and NFU, it
doesn't use any typed language, Extensionality is very simple compared to
those used in type theories, there is no restriction at all on quantifiers
in formulas, and the structure of the theory is dead simple. It is also
strictly weaker than Z, so it is safer as far as the question of
consistency of these theories is concerned.

Zuhair
```