On 3/16/2013 10:42 AM, WM wrote: > On 16 Mrz., 15:47, fom <fomJ...@nyms.net> wrote: > >> Yes, you are focused on the axiom >> of choice. > > Well that's why I raised this topic. And here are my corresponding > questions again: > > 1) Do you agree that Zermelo used a set T consisting of disjoint sets?
I would have to check on whether they were disjoint. His style in the description of intersections suggests that you are correct on this count. But, yes, the set T in the statement of the choice principle is a set of sets.
> 2) Do you agree that choosing a number from a set with more than 1 > element means writing or speaking or at least thinking the name of the > number?
No. The use of logic and axioms is justifiable as representations that formalize mathematical practice. They are normative ideals against which mathematical practice is measured.
Your question applies to the faithfulness of those representations. What is "nameable in principle" may not be materially nameable.
> 3) Only finite names can be written, said or thought.
Compact names. Finiteness in this case is supplanted by the topological terms. I can count the number of symbols. But a symbol is finite only in the sense that its mark, sound, or apperception is compact in time and, where externalized, space. With this stipulation, we agree.
> 4) There are only countably many finite names that can be written, > said or thought.
Yes. These are limitations of our material existence.
> 5) Zermelo's AC requires that uncountably many names can be written, > said or thought.
No. Zermelo's AC requires that one name can be written with certainty.
"the cartesian product of non-empty sets is non-empty"
This is why I keep returning to Robinson's "On Constrained Denotation". It is the only model-theoretic document which returns to the sense of Frege's completion of incomplete symbols. Naming populates the domain diagonal.
The domain diagonal interprets the sign of equality.
The difficulty in these matters is the nature of "in principle".
For example, another apparent limitation on human beings is our inability to not slaughter ourselves when we cannot work together. If the principles of a science are to conform with the actuality of reality, there can be no such thing as science at all. There are only the different beliefs that have so often led us to do things that most of us abhor. Although everyone likes to think relativity is a theory of time and space, what actually follows from Einstein is that the subjective a priori space and time of Kant is restricted to the subjective space and change we witness individually. The "in principle" of science is a contingent truth into which each of us must read our own subjective experience.
The presumed falsifiability that applies to empirical sciences is a confusion to many poor people without education who often rely on their faith in religious symbolism to rationalize the events of their lives.
How could they justify, as a matter of personal decision, their predominantly lawful acts on the basis of any empirical science propounded by the opulent class? The schools that produce pastors and priests also engage in logic and apply that logic to what they teach in the moral guidance they provide. Those who learn these facts measure them by their own experiences and make decisions accordingly.
Each of us must make sense of our own life first. The facts of the world are judged in relation to our personal experience. The facts of science, empirical or demonstrative, are no different. But, they can be differentiated from simple belief when they are organized "in principle".
Where your objection to "faithfulness" lies with the finiteness of feasible succession and the compactness of symbols that stand in representation of objects, mine lies only with the fact that metamathematics does not faithfully represent particular practices that seem to be required to account for uses of the sign of equality.
> 6) In this respect it resembles the statement that a second prime > number triple beyond (3, 5, 7) can be found, perhaps even infinitely > many.
My unfamiliarity with number theory, and with Diophantine problems generally, keeps me from offering an opinion on this. Were I to try, the probability that my ignorance would be displayed far outweighs the possibility of appearing intelligent.