[FOM] 600: Removing Deep Pathology 1

[Mitchell Spector]

Are these "pathological" sets still "pathological" in L, even though there's an explicit 
construction for them there?

If you answer "Yes", then the "pathology" isn't really based on non-constructivity after all.

If you answer "No", then you have to be willing to concede that, for example, the Banach-Tarski 
paradox isn't bizarre in L.

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I vote for "Yes" above, by the way.  For what it's worth, I think "pathological" is a misnomer, and 
these sets should be dubbed something like "seemingly bizarre".

To be clear, I'm thinking of such things as non-measurable sets of reals, a Hamel basis, the 
Banach-Tarski paradox, etc.

If non-constructivity isn't what's essential here (as is evidenced by L), then what's going on?

I suggest the problem may be that the bizarre objects can't be constructed in a canonical manner, 
even in L.  (An explicit construction in L would depend on a specific well-ordering of the reals in 
L, which we often think of as being canonical, in that we order things by the order in which they 
appear in the constructible hierarchy.  But this is not actually canonical -- order of appearance in 
the constructible hierarchy depends on the ordering of formulas being used as definitions in the 
L-alpha hierarchy, and in fact on your choice of details regarding the formulation of the 
first-order logic containing these formulas.  This can all be made explicit, but not in a canonical 
way.)

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One way to look at the phenomenon of bizarreness or "pathology" is in the context of the merits of 
various axiomatizations that can be proposed for mathematics.  It is customary to say that the 
merit, or lack thereof, of any collection of axioms is based not just on their self-evident nature, 
but also on the breadth of the body of consequences that they yield.

I think this is usually thought of positively ("Does this axiom system let us prove interesting 
things?"), but there's also a negative perspective: If a collection of axioms appears self-evident 
but turns out to have consequences that go against our intuition, one might question whether the 
axioms really are a suitable basis for mathematics.

We're left with the following irony: The axiom of choice, which seems self-evident, yields 
"pathological" consequences for analysis.  Yet the axiom of determinacy, which cannot be deemed 
self-evident, yields, to my knowledge, no "pathological" consequences for analysis (in spite of its 
bizarre set-theoretic consequences for cardinals greater than aleph-2).


Mitchell Spector