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Topic:
Cantor's absurdity, once again, why not?
Replies:
77
Last Post:
Mar 19, 2013 11:02 PM



fom
Posts:
1,968
Registered:
12/4/12


Re: Cantor's absurdity, once again, why not?
Posted:
Mar 15, 2013 7:09 PM


On 3/15/2013 2:12 AM, WM wrote: > On 14 Mrz., 23:36, fom <fomJ...@nyms.net> wrote: >> On 3/14/2013 5:15 PM, WM wrote: >> > >> >>> distinguishable, that means definable by finite words >> >> How does a definition "distinguish"? > > A definition is a name.
Ok. But, then I would have to ask what you mean by name. What you have replied with is "definitory" (keeping with Aristotle), but it is definitory in way that even made Aristotle uncomfortable.
Let me save some time. There are Millian names and descriptive names. And, for the purposes of mathematics, one should distinguish between descriptive names and descriptivelydefined names because one shall never ostensively indicate a physically material object as the referent of a description used in mathematics.
So, say that what you mean is a Millian name. I am making this choice based on your prior statements that would even put you at odds with the positions of Abraham Robinson in his criticism of Russellian description theory.
The sense in which you are using it is probably inadmissible because one shall never ostensively indicate a physically material object as the referent for most mathematical objects. Consequently, it has no semantics unless you are following some free logic. And, it has already been deduced that that is one of the possible ways of thinking about your claims.
But, we shall ignore this possible inadmissibility since a similar situation exists when one applies Tarski semantics to mathematics generally.
As far as names go, the modern criticism that would best seem to fit your stance is Kripke's criticism of Russellian description theory. The causal theory of naming that he gave to explain his criticism in "Naming and Necessity" would characterize your use of a name in such a way so as to say:
A definition is a authorial dubbing of an intension with a meaningful expression of the language.
So, we have intensions... http://en.wikipedia.org/wiki/Intension
...and the semiotics corresponding with observable sign vehicles. http://en.wikipedia.org/wiki/Sign_%28semiotics%29#Dyadic_signs
There is the consequent philosophical form arising from the work of Saussere and subsequent responses. http://en.wikipedia.org/wiki/Structuralism http://en.wikipedia.org/wiki/Poststructuralism http://en.wikipedia.org/wiki/Deconstruction
You probably cannot involve yourself with deconstruction. http://en.wikipedia.org/wiki/Authorial_intentionality#Poststructuralism
You probably cannot involve yourself with poststructuralism. http://en.wikipedia.org/wiki/Poststructuralism#Destabilized_meaning
So, you are left with... http://en.wikipedia.org/wiki/Structuralism
...because you certainly wish to preserve authorial intentionality... http://en.wikipedia.org/wiki/Authorial_intentionality
...as you "know reality" and others should learn it.
However, there can be problems with intentionality http://en.wikipedia.org/wiki/Intentionality#Is_Intentionality_discourse_a_problem_for_science.3F
so that you will probably end up at the Quinean double standard http://en.wikipedia.org/wiki/Indeterminacy_of_translation
Now, modern mathematics  in so far as it accepts various versions of set theory as foundational  attempts to circumvent many of these issues by claiming an extensional rather than intensional foundation.
It does not really succeed in this because it has not fully dealt with the issue of definability in terms of definite descriptions. Historically, mathematicians do not like to think about the issue of definability. But, that is a denial in the sense that the modern logical foundations will eventually force them to come to terms with those issues.
Woodin's ideas involving truth persistence under forcing (see van Frassen for truth persistence generally) are not classical ideas of truth. They are perfectly legitimate philosophically and are probably the best that any set theory using extensionality as an axiom can do.
But, your statement:
"A definition is a name"
does not quite satisfy the expectations of my question concerning how a name manages to inform "distinguishability".
> Extensionality says that sets with same elements are identical.
There are several interpretations of this statement.
Are you assuming that identity is noneliminable as on page 5 of Thomas Jech's "Set Theory" where the "standard acccount of identity" is being presumed?
http://books.google.com/books?id=pLxq0myANiEC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
Kunen also assumes a prior standard account of identity
http://books.google.com/books?id=S_ASNtsaynYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
(The standard account of identity can be found here) http://plato.stanford.edu/entries/identityrelative/#1
Are you assuming that identity is eliminable as described by Willard Quine on pages 14 and 15 of "Set Theory and Its Logic" where identity is eliminated by virtue of syntax?
http://books.google.com/books?id=S_ASNtsaynYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
If you are making this assumption, what justifies your presumption that the singular terms of your language satisfy the purport of singular reference, and, how do you justify any form of informative identity such as illustrated by "0.999...=1.000..."?
Are you assuming a metalinguistic interpretation of identity as in Zermelo's original paper?
"If two symbols, a and b, denote the same object, we write a=b, otherwise (a=b)."
Observe, however, that this does not get you out of any argument leading to "dubbings" in the sense of Kripke. Zermelo goes on to clarify the exact nature of his usage here with respect to his assertion that a singleton exists for every domain element
"..., there exists a set {a} containing a and only a as its element;"
"The question whether a=b or not is always definite, since it is equivalent to the question whether or not ae{b}."
Furthermore, observe that if you try to choose Zermelo's option here, then any naive semantics you apply will force you to address my other observations involving Leibniz' law and Cantor's intuitions concerning a nested sequence of closed subsets having vanishing diameters converging to a nonempty set having only a singleton as element. That interpretation involves an implicit use of completed infinities  one which Cantor merely made explicit.
> Therefore it must be possible, in ZF, to fix whether elements are same > or not. Therefore it must be possible to compare and to recognize > elements. For that sake you need names, unless the elements are > material objects. That's part of ZF.
In his 1908 paper, Zermelo does not say that every object of his domain has a denotation. He merely gives necessary conditions for dealing with two given denotations purporting reference to objects of the domain.
He does speak of "definiteness" and had been criticized for not being clear on this matter.
In that regard, he says,
"A question or assertion F is said to be definite if the fundamental relations of the domain, by means of the axioms and the universally valid laws of logic, determine without arbitrariness whether it holds or not."
Modern set theory has followed Skolem in clarifying the statement concerning "definiteness" (although the choices made along these lines may not necessarily be attributable to Skolem given what van Heijenoort says in his introductory remarks).
Skolem writes:
"By a definite proposition we now mean a finite expression constructed from elementary propositions of the form aeb or a=b by means of the five operations mentioned"
Those five operations to which he refers are conjunction, disjunction, negation, universal quantification, and existential quantification.
Thus, your claim concerning ZF  in so far as it corresponds with any information available to me  is inaccurate. It leads to standard forms of logic of which you and others have been critical.
By "standard forms of logic" I mean the compositionality of formal statements involving a set of primitive logical connectives necessary to conform with the grammatical requirements of a formal deductive calculus. If you wish to disagee with this, we can toss Skolem out with the rest. But, both Jech and Kunen formulate their set theory on the basis of what Skolem described.
In Jech it is on page 2
http://books.google.com/books?id=pLxq0myANiEC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
In Kunen, it is on page 3
http://www.amazon.com/IntroductionIndependenceStudiesFoundationsMathematics/dp/0444868399#reader_0444868399
One can still debate the interpretation of the quantifiers. However, it is simply irresponsible to say that something has been done incorrectly and should be ignored without providing an alternative.
An example of the appropriate way to proceed is to be found in the Markov excerpts in
news://news.giganews.com:119/qtmdnXtM3_iSVqPMnZ2dnUVZ_sadnZ2d@giganews.com
Or, one can argue concerning what it means specifically for the objects of a domain to be "definable in principle" as a way to address the relationship of names to the model theory of sets. I did this in
news://news.giganews.com:119/5bidnemPpsnq13zNnZ2dnUVZ_sOdnZ2d@giganews.com
But, like most critics, you do not believe some aspect. So, you engage in debate along the lines of an agenda rather than along the lines of a search for foundational principles.



