[FOM] The Finite Principle

[Zuhair Abdul Ghafoor Al-Johar]

This turns to be inconsistent also. The reason is that some properties that
are not shared between hereditarily finite and non hereditarily finite set
would cause inconsistencies. Example let phi be the following formula

(s)(s is ordinal & s equinumerous to z1 -> y={s})

Now for every hereditarily finite z1 the set {y|phi} is clearly hereditarily finite. But for z1 which is not well orderable this will cause {y|phi} to be
the set of all sets which is contradictive here since separation is a theorem.
Even if we say that Choice holds the problem still continues we can
pick any formula pi(s,z1) which stand for a property that do not hold for any
s for some z1 that is not hereditarily finite and then we let phi be
(pi (s,z1)->Q) such that phi hold for one hereditarily finite set when
z1 is hereditarily finite and this will cause a paradox.

To correct this principle we need to state that every subformula pi of
phi there must always exist a set satisfying pi whether parameters of pi
stand for hereditarily finite or non hereditarily finite sets which
makes the principle too complex.

Zuhair



At Thu, 15 Dec 2011 13:32:16 -0800 (PST)
Zuhair Abdul Ghafoor Al-Johar <zaljohar at yahoo.com>
wrote:
> 
> The Finite principle: if ?(y) is a formula such that
> when all its parameters stand for hereditarily finite "HF"
> sets then it follows that there exist a set {y|?(y)}that
> is hereditarily finite, then there would exist a set
> {y|?(y)}that is an element of a set as long as all
> parameters of ?(y) stand for elements of sets.
>