Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


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


Re: distinguishability  in context, according to definitions
Posted:
Feb 14, 2013 5:42 PM


On 2/14/2013 9:32 AM, Shmuel (Seymour J.) Metz wrote: > In <qImdnYCz5tRmvITMnZ2dnUVZ_oWdnZ2d@giganews.com>, on 02/11/2013 > at 10:53 AM, fom <fomJUNK@nyms.net> said: > >> Should I assume this question is contemptuous? > > More precisely, that you were slinging around symbols and words > without knowing what they mean, and inventing a local language without > providing definitions. That, by the way, is what some of the > participants in the tree thread were doing.
That last sentence speaks to the motivation for looking into "distinguishability" from a cited source in the first place.
> > You might start by thinking about what you meant by "9.999...", then > asking whether "pattern matching" has any relevance. > > Similarly, what were you trying to say when you wrote 'Now, in what > follows, the particular problem will be considering the nature of > eventually constant sequences taken to be ontologically "the same."'?
Here are descriptions of the received paradigm for use of the sign of equality  not necessarily one with which I agree, but one that I did follow in that post.
This is from Morris explaining the received paradigm with which he disagrees,
"On this account, an identity statement is a statement about the objects referred to by referring expressions with which it is made. It is not in any sense a statement about those expressions themselves. The do not figure in its truth conditions."
The received paradigm was heavily influenced by Russell, Wittgenstein, Carnap, and Quine. In large part, that means logical atomism. When representing the nature of that position, Cochhiarella writes:
"That is, an object's selfidentity or nonidentity with any other object is invariant through all possible worlds of a logical space containing that object. We must distinguish this ontological invariance from the varying semantical relation of denotation (Bedeutung) between an object and a (nonTractarian) name or definite description. The former must be accounted for within the formal ontology, the latter only within its applications."
So, in answer to your question, given two names purporting reference for one object, I took the time to clarify that the analysis was involving my best effort at navigating the transition across the syntacticsemantic ladder of a formal construct.
Carnap explains it a little better than my analysis book in college did,
"If, in a constructional system of any kind, we carry out a stepbystep construction of more and more object domains by proceeding from any set of basic objects by applying in any order the class and relation construction, then these domains, which are all in different spheres and of which each forms a domain of quasi objects relative to the preceding domain are called constructional levels. Hence, constructional levels are object spheres which are brought into a stratified order within the constructional system by constructing some of these objects on the basis of others. Here, the relativity of the concept 'quasi object,' which holds for any object on any constructional level relative to the object on the preceding level, is especially obvious."
Of course, I had to figure it out for myself the hard way.
Now, just so we understand that I am actually thinking about mathematics when I do these things, let us compare Kunen, Jech, and Zermelo 1908.
Zermelo first.
"Set theory is concerned with a domain B of individuals, which we shall call simply objects and among which are the sets. If two symbols, a and b, denote the same object we write a=b, otherwise (a=b)"
So, Zermelo is treating the sign of equality with respect to denotations, and, consequently, not according to the modern received paradigm. He did, however, recognize that more was needed,
"The question whether a=b or not is always definite since it is equivalent to the question whether or not ae{b}"
Now Jech.
It takes him a little time to get to any discussion for the sign of identity relevant to these remarks, but he begins with
"Intuitively, a set is a collection of all elements that satisfy a certain given property."
A few pages later, he simply seems to accept whatever the received paradigm for identity might be in his statement following the presentation for the axiom of extensionality:
"If X and Y have the same elements, then X=Y:
Au(ueX <> ueY) > X=Y
The converse, namely if X=Y then ueX <> ueY, is an axiom of predicate calculus. Thus we have
X=Y <> Au(ueX <> ueY)
The axiom expresses the basic idea of a set: A set is determined by its elements"
Just to be clear, Jech's last statement does not seem quite right. That characterization for a set corresponds with Frege's analogous notion of identity for concepts in abstraction of true identity for objects.
Cantor rejected that interpretation of a set in his review of Frege:
"... fails utterly to see that quantitatively the 'extension of a concept' is something wholly indeterminate; only in certain cases is the 'extension of a concept' quantitatively determined; and then to be sure, if it is finite it has a determinate number; if it is infinite, a determinate power. But, for such a quantitative determination of the 'extension of a concept' the concepts 'number' and 'power' must previously be given from the other side; it is a twisting or the correct procedure if one attempts to ground the latter concepts on the concept 'extension of a concept.'"
Or, more succinctly in correspondence with G.C. Young,
"In [remark 1 of Grundlagen], I expressly say that I only call multiplicities 'sets' if they can be conceived without contradiction as unities."
Athough you probably do not see this, Zermelo's language reflects Cantor's sentiments and Jech's does not.
Here is Kunen.
"Intuitively, x=y means that x and y are the same object. This is reflected formally in the fact that the basic properties are logically valid and need not be stated explicitly as axioms of ZFC. For example,
 x=y > Az(zex <> zey)
whereas the converse is not logically valid, although it is a theorem of ZFC since its universal closure is an axiom (Extensionality):
AxAy(Az(zex <> zey) > x=y)"
So, once again, there is deference to the received paradigm.
That post was written according to the "basic" logic that mathematicians say they are abiding by when they say they are working in ZFC in the sense of Jech or Kunen.
> Do you know what syntax is?
Not much different from noise with a little geometry.
> What do Dedekind cuts have to do with > decimal expansions, other than the trivial sense that they can be used > to construct a model of the Reals?
Symmetry.
In the hierarchical constructions of the reals, the identity of a real number within the system is achieved by retaining the order relations from each step up the ladder beginning with the simple sequential ordering of the natural numbers. With the Dedekind cuts, one must decide on whether the construction shall be based upon least upper bounds or greatest lower bounds. I recognized a possibility to investigate the question in terms of a choice among two equivalent names for the same object whose indistinguishability rested on the presumption of a completed infinity.



