fom
Posts:
1,968
Registered:
12/4/12
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Re: Matheology § 223: AC and AMS
Posted:
Mar 16, 2013 1:17 PM
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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.
Another human limitation.
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