On 2/20/2014 5:39 AM, email@example.com wrote: > > > Zeitgeist on the odds of my students to understand transfinite set theory:" if they can be convinced that what their senses tell them may be Not be the whole picture, then they may have a chance." > > Virgil, appearing as Wisely Non-Theist: "There are 'more' real numbers than there are finite definitions to define them with, so most reals can only be defined collectively, not individually." > > Ben Bacarrisse emphasized: "They are not 'entire undefinable'. The set of them can be defined." And he added: "You can know things about the set. For example, that it can't be bijected with N." > > William P. Hughes: "A subcollection of a listable collection may not be listable." > > Alan Smaill: "After all, since matheology accepts undefinable real numbers, then why are you trying to suggest that it does not accept undefinable enumerations?" > > That is true. I never got a grasp of this idea: Why should undefinable definable enumerations (aka lists) be exemptet from the list of unlisted exemptions? > > Regards, WM >
I am not entirely certain of your question. But, I will give you two answers. Here is the first.
Consider, for a moment, the origins of the problem.
Whatever deficiencies may have existed in Euclid, magnitudes and multitudes had been treated separately. Both had been related to the notion of "part", and, the definition of "point" had been based on the negative criterion of having no parts.
Admitting my ignorance of non-Western mathematics, suddenly Viete and Descartes appear. Viete conflates magnitudes and multitudes with his algebra. Descartes conflates magnitudes and multitudes with his analytic geometry.
Now, jump ahead to Bolzano. Bolzano had a certain aesthetic for logical principles. And, seemingly unrelated, he devised a proof for the intermediate value theorem specifically directed at eliminating geometric intuition from the explanation of how algebraically defined curves intersected.
The proof involves a fixed point argument involving two continuous functions over a compact set. Because of the presumed boundary conditions, there must be some point on the interior of the set at which the two functions have the same value.
Now, between Bolzano and Cantor, functions had been generalized. I believe that this is attributed to Dirichlet. With Cantor and Peano, it is recognized that notions such as dimension fail to be invariant with respect to this definition of function. Only later, when topology becomes organized, is it recognized that continuity is an essential aspect to the notion of dimension invariance.
To the chagrin of many, including yourself, Cantor argues successfully for a metaphyical existence of limits as numbers and for a transfinite arithmetic. He is supported in this by the Hilbert school. But, because of contradictions arising from the inexactness of natural language, the effort to rescue set theory leads to a heavy formalization of mathematics at the hands of logicians.
In the midst of this, predicative mathematics arises. Under the presumption that "objects" are primary, the formulation of set-based mathematics becomes dominated by a typed hierarchy in which functions do not arise from definitions in the sense of Bolzano. To the contrary, they correspond with their extensional definition in some typed structure.
So, whether or not one may "speak of a function, a relation, or a listing" depends upon whether or not it has an extensional definition in a definite interpretation of the language. If one cannot, then it is undefinable (relative to a model). What is generally true, on the basis of Cantor's diagonal proof, is that it is not possible to have a model in which everything is "definable simpliciter".
I started this little story with Viete and Descartes because the abstract treatment of languages without interpretation presents itself in universal algebra and the problem of associating every geometric point with a name begins with analytic geometry.
But, the strict constraints arising from predicative mathematics is grounded in the very kinds of objections which you, yourself, make. And, that is the very perspective which demands that functions are logically posterior to objects. This "defeats" Bolzano's proof in the sense that the names of the real numbers are prior to the two functions upon which his proof relies.