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.