Naive explanation
The n+1_ary relation ∅ (y,x1,..,xn) is to be understood as a rule linking objects substituting variables x1,...,xn which are called here as input variables to objects substituting variable y which is called the output variable.
Now ∅ would be called "closed over S" where S is some class, if and only if when all its inputs comes from S (i.e. are elements of S) it outputs objects all of which are in S also.
Formally closure to be defined as:
∅ is closed over S ⇔ ∀x1..xn ∈ S [∃y ∅(y,x1,..,xn) ∧ ∀y (∅(y,x1,..,xn) ⇒ y ∈ S)]
This closure notion is well known, for example we'd say that the 3_ary relation "summation" is closed over the set N of all naturals, because for every two input naturals x,y the output x+y is also a natural; while the 3_ary relation "subtraction" is clearly not closed over N.
We are interested in relations that close over the set HF of all hereditarily finite sets for the purpose of generalizing them to be closed over the whole class of all sets.
Of course not every n_ary relation ∅ is closed over HF, for example the relation ∅(y,x1) defined as "x1 ∈ y" which is a rule that links each input object (i.e.,substituting x1) into every output object (i.e.,substituting y) that contains that input object as an element, clearly it is not the case that the output of ∅ would be always hereditarily finite for every hereditarily finite input, since for example ∅ does link the empty set (input) to HF (output) and the later is clearly not hereditarily finite although the former is.
Now I'll pose what I'd label as the "naive generalization principle" as:
"IF ∅ is closed over HF, Then ∅ is closed over V".
Where V is the class of all sets.
Now this naive generalization principle is inconsistent!
To show that, I'll present two counter-examples:
Example i: Let ∅(y,x1) be definable after the following formula
∀z (z ∈ y ⇔ z ∈ V ∧ ∃uw (u ∈ x1 ∧ u is infinite ∧ u C w ∧ z ∈ w))
where u C w is the subclass relation defined as: "∀m (m ∈ u ⇒ m ∈ w)".
Now clearly y is the union of all supersets of infinite elements of x1.
Now every hereditarily finite object substituting x1 would be linked by ∅ to the empty set as output (i.e. y is always empty if x1 ∈ HF), which is hereditarily finite, so ∅ is closed over HF. Now by naive generalization, it would follow that ∅ is closed over V, i.e. we'll be having:
∀x1 ∈ V [∃y ∅(y,x1) ∧ ∀y (∅(y,x1) ⇒ y ∈ V)]
which is paradoxical because let x1 be the set {HF} (i.e. the singleton set whose sole element is the set HF), then clearly y would be the set union all supersets of HF, which is simply V itself, so we'll have V∈ V and this is paradoxical (because we do have separation over sets as a theorem).
Example ii:
Take ∅(y,x1) to be definable after the following formula:
∀z (z ∈ y ⇔ z ∈ V ∧ ∀s (s is finite ∧ s=x1 ⇒ z=s))
Now clearly ∅ is closed over HF, since for every hereditarily finite set input x1 the output y is {x1} which is hereditarily finite also.
Now from naive generalization we'll have ∅ being closed over V, which is paradoxical since it would also lead to V ∈ V (just simply input HF).
To analyze those two pathological examples, we saw that in the first example there was a pathological subformula which is "u is infinite ∧ u ∈ x1" whereby that was bypassed by all hereditarily finite inputs because it was never fulfilled (i.e. does't have an output), but once we go outside the realm of HF some objects containing infinite elements would be trapped by this formula because it would be FULFILLED (i.e. has an output) and that lead to feeding the formula u C w leading to y containing all elements of supersets of u as elements of thus y being V itself.
In the second example the opposite happened, the pathological subformula was "s is finite ∧ s=x1" where the "non fulfillment" of which caused the pathology since it is the antecedent of the formula and it becoming false would allow every z in the conclusion to fulfill the formula thus pumping up y to V. Now this was bypassed by all hereditarily finite inputs because this pathological formula was ALWAYS fulfilled.
I refer to the above two situations as:
i. Pathology by fulfillment.
ii. Pathology by non fulfillment.
I think that there are no other kinds of pathology, and on that assumption I propose the following remedy:
For ∅ to be eligible for the generalization principle it must be definable after a formula phi such that every subformula pi of phi: if pi is ALWAYS fulfilled (i.e. has an output) when the inputs are hereditarily finite sets, then it should be ALWAYS fulfilled for all set inputs; and if pi is ALWAYS denied (i.e. doesn't have an output) when the inputs are hereditarily finite sets, then it should be ALWAYS denied for all set inputs.
If the above condition was met then we say that ∅ is existentially equivalent to itself, and we denote this as: ∅ ⇐∃⇒ ∅, this is formally defined as:
∅ ⇐∃⇒ ∅ if and only if for every subformula ψ of ∅ the followings are met:
∀x1..xn ∈ HF ∃y ψ(y,x1,..,xn) ⇒ ∀x1..xn ∈ V ∃y ψ(y,x1,..,xn)
∀x1..xn ∈ HF ~∃y ψ(y,x1,..,xn) ⇒ ∀x1..xn ∈ V ~∃y ψ(y,x1,..,xn)
for all possible orders of quantification.
However it might be reasonable to fix the order of quantification over variables in each tested subformula ψ of ∅ to be the same as the order of quantification over them in the main formula ∅.
So the corrected generalization principle would read:
"IF ∅ ⇐∃⇒ ∅ and ∅ is closed over HF, Then ∅ is closed over V".
Now if this principle is true, then it would enable us to derive natural axioms for set theory and thereby guide us develop some natural extensions of set theory.
I think the above axiom is of the strength of ZF (since Infinity is implicitly assumed here by declaring HF being a set).
So formally speaking this principle coupled with Extensionality, impredicative class comprehension of Morse-Kelley, and axiom of existence of a power-inductive set (a set containing the empty set among its elements, that is closed under existence of power sets), and of course axiom of infinity, would lead to a theory that can interpret all axioms of ZF.
Zuhair Al-Johar
December 4,2015