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Topic: Notation
Replies: 14   Last Post: May 27, 2013 1:46 AM

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Posts: 1,968
Registered: 12/4/12
Re: Notation
Posted: May 26, 2013 5:42 PM
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On 5/26/2013 2:06 PM, Ross A. Finlayson wrote:
> On May 26, 11:25 am, fom <> wrote:
>> On 5/25/2013 8:36 PM, William Elliot wrote:

>>> On Sat, 25 May 2013, fom wrote:
>>>> On 5/24/2013 10:57 PM, William Elliot wrote:
>>>>> V is the ZFC universe.
>>>>> L is the constructible universe.
>>>>> L_omega0 is the omega_0th level of the constructible universe.

>>>>> Correct or needing correcting?
>>>> Your apparently simple question seems to have generated
>>>> some interesting replies.

>>> Seemingly off the mark because I'm asking about notation and not about
>>> theory.

>> Just to clarify something concerning your distinction here,
>> let me observe that the sign of equality in "V=L" is
>> metamathematical or metalogical or metalinguistic or
>> however some purist would like to take it as not being
>> part of the "object language".
>> In other words, part of the reason your question received
>> "theory" replies is because of the class/set distinction
>> and its model-theoretic implications.
>> These particular notations correspond precisely with
>> the point at which philosophy and mathematics (prior
>> to intuitionism and categories) meet. Enough said.

> What part of philosophy has non-sets...

To be honest, Ross, most of the mathematical words
involved with these topics have become meaningless to

Here is simplicity.

In the link,

you will find the statement,

"The priority monist will hold that the one basic concrete
object is the world (the maximal concrete whole). She will
allow that the world has proper parts, but hold that the
whole is basic and the parts are derivative."

In Chapter 6 of "Logic, Logic, Logic" by Boolos you will
find the statement,

"A different conception of set, to be examined below,
is the doctrine of 'limitation of size'. The doctrine
comes in at least two version: On a stronger version
of limitation of size, objects for a set if and only
if the are not in one-one correspondence with all the
objects there are. On a weaker, [...]"

So, the question becomes how to differentiate "all
the objects that are" from sets. Well, that happens
to be the same question as how to interpret the
universal quantifier in set theory as an *ontologically
foundational* theory.

Referring again to the link,

you will find the designation,

"Ontological priority: there is an irreflexive,
asymmetric, and transitive relation of ontological
priority between entities."

So, it is ludicrous to accept the standard account
of identity based on logical atomism. In relation
algebra, there are 4 necessary relations: full, empty,
identity, and diversity. Combining the roles of
"diversity" and "ontological priority" I can grammatically
define a relation,

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

that is provably irreflexive, asymmetric, and transitive
within the standard deductive calculus and without any
interpretative philosophy beyond the fact that the
transformation rules of the calculus "mean what they do".

Referring again to the link,

one has the designations,

"Ontological foundationalism: the ontological priority
relation is lower-bounded."

"Concreta foundationalism: there are lower-bounded
ontological priority relations within the domain of
concrete objects."

The grammatical definition,

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

provides for the directionality to establish
these principles. But, it is the complexity
associated with the identity relation that must
be addressed before these two aspects of priority
monism can be properly implemented.

I have written a set theory in which the only
proper class is the universe (Boolos' strong
limitation of size). The null class is the
ontological lower bound. But, the singletons
are the concreta. It may be compared with
Euclid. On the one hand, "a point is that
which has no part" describes the empty
set. On the other hand, "a unit is that
by which what exists is one" describes
the relation of singletons to singular
terms in the universe.

The subtle complexity of "identity" and
"diversity" come into play because singular
reference to the universal class follows
from the role of description theory in
the history of foundations. But, Hilbert's
formalism obscured this relationship and
the influence of Russell and Tarski divorced
"names" from foundations as an extra-logical

So, a simple beginning becomes an almost
impenetrable argument.

In the end, the description theory trail leads
to David Boersema's "Pragmatism and Reference"
in which he points out that "the given" (for
pragmatic philosophers) is a given whole.
Prior to individuation, the given whole must
be interpreted as decomposed into parts.
Only then may the identity presuppositions of
description theories apply.

You may contrast that idea with the distinction
between "priority monism" and "priority pluralism"
in the often-referenced link,

I have posted my formal sentences several
times. You can find them in the strangely
titled post,


Let me add that I did what I was supposed to do.

That is, when faced with an independent question,
one must come up with reasonable philosophical
positions for the introduction of new axioms. It
appears, however, that my efforts are in vain.
Half of mathematicians believe that God gave
Kronecker the natural numbers. Half of mathematicians
believe that God gave Russell an intractable paradox.
And Hilbert ensured that skeptics could make any
argument meaningless in order to save Cantor's

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