[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.

> 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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