Date: Dec 31, 2012 3:21 AM
Author: fom
Subject: Re: Freiling's axiom of symmetry -- alternate intuitive considerations



Because the class theory is based on sentences
having self-referencing syntax,


Definition of proper part:
AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

Definition of membership:
AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))



some terminology and some further explanation
is in order.



=================
Only symbols intended to represent relations
may be introduced into the language by such means.
Since such language symbols cannot be eliminated
through substitutions, they are collectively
referred to as being "morphemic." Among these
morphemic symbols, there can be only one self-defining
symbol and the term "primitive" is reserved for
reference to this symbol.

Symbols that are not morphemic are derived.

The notion of "definition" used here is non-standard.
It conveys the idea that a formal language intended
for foundational investigations must be the product
of synthesis rather than the application of analysis.
The sentences above have syntax appropriate to be
the first two formulas of any derivation using the
symbols defined by that syntax. Symbols intended
to name objects are presumed to extend the language
through descriptions whose uniqueness is provable.
In this regard, such a symbol can be a canonical
selection from among terms. Thus, because


null()



is a provably unique denotation that can generate


S(null())

S(S(null()))



in combination with the successor function, S(),
the terms,


S(null())

S(S(null()))



can be taken to be canonical denotations for which
descriptions could be formed provided that S()
satisfies the usual characterization of a function.



=================
The sense of the definitions given above can
be compared with two axioms found in Rosser's
"Simplified Independence Proofs".

First, recognize that the universal subformulas
given by


Az(ycz -> xcz)

Az(ycz -> xez)



have simple pictoral representations relative
to concentric circles having increasing diameters.

It is the representation for the existential
subformulas in which we are interested for
comparison.

Although there is a degenerate case, the non-degenerate
representations for these formulas


Ez(xcz /\ -ycz)

Ez(xez /\ -ycz)



positions the subset or element in the intersection
of y and z relative to a non-trivial symmetric
difference.



Rosser's second axiom reads:

"If xeX is in each of two basis sets B_1 and
B_2, then there is a basis B_3 such that
(B_3 c= (B_1 n B_2))"


The definition,


Definition of proper part:
AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))




is simply asserting that "to be a subclass, a class
must be interpretable as a basis element in a
topology on its superclass".



Rosser's first axiom reads:

"For each point xeX there is a basis set B
such that xeB"



The definition,


Definition of membership:
AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))




is simply asserting that "to be an element, a
class must be interpretable relative to the
basis element with which it would be associated
in a discrete topology on the superclass of
that basis element".



=================
Since the definition of morphemic predicates
would appear to present the possibility
of vicious circle paradoxes, the following
model, although in violation of any expectation
with regard to the axiom of regularity, conveys
a sense that these definitions do not lead
to paradox.

Among the natural numbers having no squared
elements in their prime decompositions, one may
interpret


AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))



as "x is a proper divisor of y"
and


AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))



may be interpreted as "x is a prime
divisor of y".



So, 5 is a proper divisor of 35 because,
according to the existence assertion of
the axiom, it is also a proper divisor
of 55.

To actually force the interpretation of xey
to be different from xcy requires more


AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) -> Az((xez /\ -ycz) ->
(-(Ew(wey /\ -xew) <-> Aw(xcw <-> ycw)))))



So, 3 is a prime divisor of 6 has
2 is a prime divisor of 6 but not
3 is a prime divisor of 2.

But, 3 is a prime divisor of 3, when
obscured by different denoting
symbols, must be identified with
as atomic in relation to the order.

With this model, however, there are
not the complexities of denotations,
demonstratives, and descriptions which
justify leaving the set-theoretic
formulations ambiguous.