The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 2,720
Registered: 2/15/09
Re: Notation
Posted: May 26, 2013 11:01 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On May 26, 2:42 pm, fom <> wrote:
> 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
> me.
> 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
> concept.
> 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,
> news://
> 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
> paradise.

Monisms as to Boole: Boole basically wrote the universe or context as
1, then indicated how we might write V \ x as 1 - x. So we might look
to the Boolean as simple for the monist, or as you note to Aristotle
for the genera before species, where set theory (as is modern) is of
the cumulative hierarchy, as it were.

For that matter than we might look to Boole and De Morgan and Hamilton
as to what is generally introduced as logic, in syllogistic logic, and
analytic, then as to set theory as at its root constructivist, and

Then, a notion of a combination of principles (or reduction from
synthetic constructions to analytic principles) is seen in quite the
most general, here we might look to theories of types as built on set
theory (eg categories in using modern mathematics) as an example of
this kind of conciliation.

Nice of you to introduce Boolos, a 20th centory logician who may well
have already covered the desirable features of theory with the
acknowledgment of the limitations of either the purely synthetic or
analytic, and the necessity of their combination of the total and

Then, of note is a reference to Boolos' rehabilitation of Frege's
Grundgesetze, here in terms of the action in philosophy to have the
technically analytic result in the utility of the synthetic for
philosophically logical and logically philosphic foundations for
mathematics, there's an example of such technical underpinnings as
might be of note and use to represent a suitably concise form for
discussion on the general matter: general matters.

Far from it from that to be said to accomplish this goal, along the
lines of the Hilbert program is not to complete mathematics then to
integrate for completion the theoretical foundations of mathematics in
as to their completion, yet, that is the statement of what would see
in the basically analytic and synthetic as to, for example, an
axiomless system of natural deduction, for inference, induction, and
reason. This is synthetically from null or void, and analytically:
that the first principles are final causes, for mathematics as a

I'm reminded of your earliest posts, fom, and their plainly
enumerative content: from what simplest principles and observation,
is thus reason evidenced? In describing equality as tautology and
identity, where is it that what are otherwise seen as the simplest
logical primitives are themselves structured, and, how is it so that
then: this truth-preserving theory encompasses "no paradox" and "all
and none"?

That would be of general interest, particularly in the general
assignment of an integrated logic with general bibliography, in as to
the parallel carriage of fact from the various to and from the
specific and macro and micro and synthetic, and analytic.

Basically, logic is reversible in: that the complement: is itself.

And the infinite is infinite.


Ross Finlayson

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.