[FOM] Removing Deep Pathology 1
joeshipman at aol.com
joeshipman at aol.com
Wed Aug 19 01:48:07 EDT 2015
Further observations:
My last question below was rhetorical, a partition of the unit square into a countable union of graphs and cographs gives a set which is countable on vertical lines and co-countable on horizontal lines, whose iterated integrals exist and disagree, a situation which I showed in my thesis (and Harvey had earlier showed) cannot be proven to occur in ZFC. This is a very deep kind of pathology (you can prove it trivially with CH and Choice, but Choice alone won't do it, it's consistent that iterated integrals of non-negative functions never disagree in any dimension).
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).
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).
Hendrik says that partial noncomputable functions aren't pathological but total ones are -- what about the Busy Beaver function?
Going back to Harvey's original post: the Shoenfield Absoluteness Theorem and related results guarantee that for almost all the theorems we care about, monsters may be ignored.
-- JS
-----Original Message-----
From: Joe Shipman <joeshipman at aol.com>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Cc: W.Taylor <W.Taylor at math.canterbury.ac.nz>
Sent: Tue, Aug 18, 2015 3:49 pm
Subject: Re: [FOM] Removing Deep Pathology 1
1) and 2) are extremely shallow pathology because their definitions are
practically trivial in terms of binary expansions and require no choice.
3),
4), 4a) obviously follow from a choice function on the reals, moderately
pathological and equivalent to the original f(x+y)=f(x)+f(y) in that
respect.
3a) is obvious from a cardinality argument without choice, mildly
pathological because of the difficulty of defining one
6) Once you understand
computation, not pathological at all, but if you are speaking informally and not
familiar with precise definitions, slightly pathological.
7) Trivial without
needing choice for any notion of "definable" that entails "specifiable by a
finite string in a finite alphabet", no more pathological than "undefinable real
number" because the concept of "determined by a finite amount of information" is
very intuitive despite the difficulties of formalizing it.
8) trivial if you
accept "order type" as a coherent notion, lots of nice examples exist, a set of
equivalence classes is a much more fundamental notion than objects selecting one
representative from each equivalence class.
5) seems the most deeply
pathological, since it's only obvious if you assume CH. How do you prove it in
ZFC?
Sent from my iPhone
On Aug 18, 2015, at 2:05 AM,
W.Taylor at math.canterbury.ac.nz wrote:
>> There is an aspect of mathematics
that I call
>>
>> *deeply pathological*
>
> I like the f(x+y) = f(x) +
f(y) example, as a case of deep pathology.
>
> May I ask the list, for
clarification, where the following might come
> on the spectrum from
"superficial pathology" to "deep pathology".
>
> 1) The Cantor subset of
[0,1]
>
> 2) Peano's space-filling curve.
>
> 3) The existence of a
non-measurable subset of [0,1]
> 3a) " " " " non-Borel "
" "
>
> 4) The partition of 3-space by (infinitely long) non-parallel
lines.
> 4a) " " " " " non-parallel circles.
>
> 5) The
partition of [0,1]^2 by countably many graphs and co-graphs.
>
> 6) The
existence of a non-computable function (from N to N).
>
> 7) The existence
of a non-definable (countable) ordinal.
>
> 8) The existence of an
uncountable ordinal.
>
> Thanks in advance for all clarifications.
>
>
Bill Taylor
>
>
>
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