fom
Posts:
1,969
Registered:
12/4/12


Re: Ordinals describable by a finite string of symbols
Posted:
Jul 7, 2013 8:29 PM


On 7/7/2013 5:23 PM, Julio Di Egidio wrote: > "fom" <fomJUNK@nyms.net> wrote in message > news:H76dnfhczfi8WETMnZ2dnUVZ_judnZ2d@giganews.com... >> On 7/7/2013 8:02 AM, Julio Di Egidio wrote: >>> >>> I have tried and reread the thread: I fail to see all the connections, >>> but that's good. :) >> >> It is in Kant that one finds precisely the >> remark that compares the relationship of a >> universal judgement to a singular judgement >> as being that of infinity to unity. >> >> Apoorv attaches the introduction of infinity >> into mathematics as related to the universal >> quantifier. >> >> Not grasping the sense of the quote (which is >> fine  who wants to read this anyway?), he >> asked additional questions that introduce even >> greater complexity. >> >> I believe you have read "Tractatus LogicoPhilosophicus" > > Yes, of course, and even studied a bit about it: I just have zero > memory, but for the bottom lines... > >> Wittgenstein rejects Leibniz' principle of identity >> of indiscernibles and proposes that the sign of >> equality is eliminable on the basis of each object >> having a unique name. > > Wittgenstein also rejects Russellian theory of definite descriptions (if > that's the exact name).
Yes. Descriptions individuate according to the principle of identity of indiscernibles.
Here is a link to "the standard account of identity".
http://plato.stanford.edu/entries/identityrelative/#1
You will find that the indiscernibility of identicals is included. Its converse  the identity of indiscernibles  is not.
So, any "naming" done with descriptions is not part of the "standard account". Now, with respect to foundations, note that both Jech and Kunen defer to the theory of identity of firstorder logic when discussing the axiom of extension. Thus, in these theories, identity is not eliminable.
What is being emphasized in this conception is the ontological interpretation of 'x=x'  that is, objects are selfidentical.
There is, however, a semantic interpretation of 'x=x' wherein each occurrence of 'x' is consistently interpreted. This may be compared with Wittgenstein's idea of eliminability of identity. However, this is merely the result of the ontological interpretation.
What Wittgenstein had been attempting to eliminate is informative identity,
'x=y'
Where this is at odds with other views is that it delineates two forms of informative identity. There is stipulative informative identity where one is asserting that 'x' and 'y' will be given the same interpretation. Then, there is epistemic informative identity where one must determine if 'x' and 'y' can be given the same interpretation.
It is the epistemic informative identity which is associated with descriptions and proofs that a description does, in fact, delineate a single object of the domain. You can find this discussed in Morris' "Understanding Identity Statements".
In his "Comments on Sense and Reference", Frege does a nice job of considering the uses of identity statements. Other than that, I have only found any significant discussion in Aristotle. Most of that is in "Topics" (I look at "Topics" as little as possible because its primary focus is rhetorical argument from beliefs.).
> For the Wittgenstein of TLP the world is the > totality of facts, but these are the facts of language! >
I always seemed to think that he had been attempting to claim logic as an ideal language in lock step with reality.
>> Modern model theory "extends languages" with constants >> without definitions. This is an algebraic approach (in >> the sense of universal algebra), and, is thus relevant >> to the perspectives of Skolem. > > Well, I have no qualms with an algebraic development of mathematics, but > not for foundations: no "primitive" is purely syntactical when the > language is a language; equivalently, no proof or argument is ultimately > purely syntactical. >
That is one of the questions for apoorv in all of this. He attempted to use Skolem's views to reduce to a countable model of set theory. But, those views derive from an algebraic, firstorder view.
That puts his desire to restrict on the basis of 'nameability' at odds with the mathematics which guarantees a countable model.
>> But, it is in the nature of definitions that symbols are >> introduced with respect to properties. This is an application >> of Leibniz' principle of identity of indiscernibles. > > Indeed, going round in circles. >
I know. I have focused on the model theory of set theory for a long, long time.
If you think about my primitives (I prefer the term morphemic relations),
AxAy(xcy <> (Az(ycz > xcz) /\ Ez(xcz /\ ycz)))
AxAy(xey <> (Az(ycz > xez) /\ Ez(xez /\ ycz)))
you will realize that the "foundational status" is recognizable by "circular definition" which, in fact, is the least complex syntax from an "infinite schema" of axioms obtained by substitutions into themselves.
These sentences are consistent with Aristotelian "immediate principles" in his theory of demonstration.
For the most part, this idea is foreign to modern viewpoints.
>> So one is confronted with the relationship of uninterpreted >> symbols of a language as treated in model theory and the >> defined symbols which Apoorv hopes to use to reasonably >> restrict notions in set theory. >> >> Does that help? >> >> It just gets worse... > > Well, I'm loving it. I just wish I could offer more precise quotes and > references, as I guess you'd appreciate it, but, hey, we do what we > can... :) >
Isn't one parrot enough?
:)

