[FOM] Removing Deep Pathology 1
Joe Shipman
joeshipman at aol.com
Tue Aug 18 15:49:39 EDT 2015
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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