[FOM] Removing Deep Pathology 1
Hendrik Boom
hendrik at topoi.pooq.com
Wed Aug 19 12:05:06 EDT 2015
On Wed, Aug 19, 2015 at 01:48:07AM -0400, joeshipman at aol.com wrote:
...
>
> Some other levels of pathology not represented below include:
>
>
> Banach-Tarski sets (logically similar to nonmeasurable sets but much more surprising because they involve only finitely many sets in the decompositions) are arguably more pathological than nonlinear f(x+y)=f(x)+f(y), but less pathological than the anti-Fubini set in example 5).
Those werre the first examples that gave me some doubts about the axiom
of choice.
>
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> The existence of a nonprincipal ultrafilter on N is less pathological than the examples which use full choice on sets of reals (not only is a weaker form of Choice necessary, but the SPACE of ultrafilters is the first nontrivial example of a very standard construction, the Stone-Cech compactification, and a single element of such a standard object can't be very objectionable.
>
>
> Hendrik Boom seems to imply that there are explicit constructions of
> partitions of space into nonparallel lines or nonparallel circles--if
> there are, I'd like very much to see them. "Pathological" such
> partitions are trivial to construct with a well-ordering of the reals
> (because any line or circle intersects any other one in only finitely
> many points, so if you give the reals a well-ordering such that each
> element has less than continuum-many predecessors there will always be
> room to find a new line or circle through a new point and disjoint
> from previous lines or circles).
My mistake. I was thinking about the Hopf fibration, which fills space
with circles, altough one is of infinite radius, but I suspect it fails
your "nonparallel" clause, which I still find unclear. What *does*
parallel mean for circles?
I have no such example for straight lines.
>
>
> Hendrik says that partial noncomputable functions aren't pathological
but total ones are -- what about the Busy Beaver function?
Well, as a constructivist, I can't really prove it's total. If I could,
it would be computable.
-- hendrik
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