[FOM] 602: Removing Deep Pathology 2

Sun Aug 23 18:35:38 EDT 2015

Harvey, what is the strongest form of the axiom of choice you are familiar with such that the kind of "deep pathology" that you talk about eliminating cannot be proven in ZF to follow from it?

-- JS

Sent from my iPhone

> On Aug 22, 2015, at 6:42 PM, Harvey Friedman <hmflogic at gmail.com> wrote:
> 
> Continuing from http://www.cs.nyu.edu/pipermail/fom/2015-August/018886.html
> 
> 2.1. Brief replies to responses.
> 2.2. Robustness of Borel.
> 
> 2.1. BRIEF REPLIES TO RESPONSES
> 
> From Walter Taylor:
> 
> 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.
> 
> REPONSE: I am focusing on the status of individual objects in terms of
> the pathology scale. The pathology scale is not at all social, but an
> objective mathematical construct. It links with the social when one
> considers what research to support with what resources.
> 
> And in particular I have enough to do to deal with the specific kind
> of pathology which is the deepest pathology of individual objects
> general familiar to mathematicians. It is also important to link this
> up to spaces of objects.
> 
> Here is a useful distinction that just came to me.
> 
> A. Individual deep pathology.
> B. Ensemble deep pathology.
> 
> In individual deep pathology, we have the failure to provide a
> definition of the object in the usual language for mathematics. This
> is a profound LANGUAGE BARRIER. This Language Barrier puts the kind of
> deep pathology familiar to mathematicians in a CATEGORY BY ITSELF. The
> LANGUAGE BARRIER is sacred.
> 
> E.g., we know using ZFC that there exist nonlinear solutions to f(x+y)
> = f(x) + f(y). But can you give an example? How about an example even
> with 100 pages of complicated description in some sort of acceptable
> mathematical form?
> 
> THEOREM. There is no formula in the language of set theory such that
> ZFC proves defines a discontinuous solution to f(x+y) = f(x) + f(y).
> Furthermore, this is true even with parameters for real numbers in the
> following sense. There is no formula in the language of set theory,
> with letters for parameters, such that ZFC proves that the parameters
> can be assigned real numbers so that the expression defines a
> discontinuous solution to f(x+y) = f(x) + f(y).
> 
> The second sentence above immediately leads to the question of just
> what parameter space can be used for this negative result.
> 
> This is in a category of deep pathology totally unto itself, totally
> transcending any specific, however complicated, mathematical
> construction. This is because there just cannot be any mathematical
> construction. The Language Barrier is sacred.
> 
> In Ensemble deep pathology, one typically has a completely acceptable
> totally simple description of a family of mathematical objects, where
> we know that there are good elements and there are pathological
> elements. This is the case with the ensemble of all solutions to
> f(x+y) = f(x) + f(y). The good ones are the ones f(x) = cx. Later in
> this posting, we want to make the crucial definitions here so that
> this phenomena can be systematically studied. In particular, we want
> to REMOVE the bad part of this ensemble, being left with the GOOD
> PART, consisting of the f(x) = cx.
> 
> In Taylor's list, only 3 (not 3a), and 5, exhibit this kind of deep
> pathology. I don't know the exact formulation of 7).
> 
> Following Shipman
> http://www.cs.nyu.edu/pipermail/fom/2015-August/018896.html Banach
> Tarsi sets and non principal ultrafilters on N are very much in this
> deep pathology category I am talking about. One has the exact analog
> of the above Theorem for these as well.
> 
> From McLarty http://www.cs.nyu.edu/pipermail/fom/2015-August/018896.html
> I gather that these fibration statements are NOT the kind of deep
> pathology I am talking about here. Perhaps there are variants of these
> kinds  of statements that are.
> 
> Martin Davis http://www.cs.nyu.edu/pipermail/fom/2015-August/018904.html :
> 
> "Everywhere continuous nowhere differentiable functions were once regarded
> as pathological. At least since fractals have entered mathematical
> discourse, they are commonplace.
> 
> I suspect that the concept of "pathological" is social rather than
> mathematical."
> 
> Since I am talking about the LANGUAGE BARRIER, everywhere continuous
> nowhere differentiable functions and the most horribly complicated
> fractals and much more bizarre creatures are "beautifully tame" in
> comparison. And because this is a fundamental LANGUAGE BARRIER, the
> situation simply cannot change over time.
> 
> OK, well, maybe the entire Milky Way galaxy will go supernova next
> month. What will then happen to the September FOM Archives?
> 
> Very powerful hard nosed Theorems like the above tell us that I am not
> talking about anything that is social. It only becomes social when one
> gets involved in whether people should be rewarded for research in
> certain directions, and in what way, etcetera. That's not what I am
> talking about here.
> 
> CAUTION: I am NOT ruling out the possibility that the study of deeply
> pathology objects or deeply pathological ensembles without having
> REMOVED(!) the pathological part, just might lead to some insights
> into the Non Deeply Pathological. Two comments.
> 
> i. This is likely to be rare, even rarer if objective people who do
> not have a stake in the deeply pathological see if the use is in some
> sense fake.
> ii. Even if it occurs, it does not reflect on any intrinsic
> mathematical interest of this kind of deep pathology.
> iii. An analyst colleague of mine has gotten interested in i), whether
> the kind of deep pathology under discussion has ever really played a
> role in understanding the well behaved. His preliminary idea is that
> there are no such real examples - the deep pathology can be
> systematically removed. (My Incompleteness work has nothing to do with
> the kind of deep pathology we are talking about here, but rather
> another kind of "wild thing" that is not familiar to mathematicians
> generally (at least not yet).
> 
> 2.2. ROBUSTNESS OF BOREL
> 
> DEFINITION. A Polish space is a complete separable metric space. Let T
> be a Polish space. The Borel subsets of T form the least sigma algebra
> containing the open subsets of T.
> 
> In this section, S,T always denote Polish spaces. Products, even
> countably infinite products, of Polish spaces are Polish spaces with
> the standard product metric.
> 
> THEOREM. Every uncountable Polish space T contains a perfect set (a
> compact set S where every point in S is a limit point of S). T and its
> perfect sets have cardinality c.
> 
> BOREL ROBUSTNESS. Let S,T be Polish spaces of the same cardinality.
> There is a bijection from S onto T which
> i. the forward image of every Borel set in T is a Borel set in T .
> ii. the inverse image of every Borel set in T is a Borel set in T.
> 
> DEFINITION. Let F:S into T. F is a Borel isomorphism if and only if
> i,ii hold above. F is Borel if and only if the inverse image under F
> of any Borel in T is a Borel set in S.
> 
> THEOREM. Every Borel bijection from S onto T is a Borel isomorphism
> from S onto T.
> 
> THEOREM. F:S into T. The following are equivalent.
> i. F is Borel.
> ii. The graph of F is a Borel subset of S x T.
> iii. The inverse image under F of any open set in T is Borel in S.
> iv. The inverse image under F os any open ball in T is Borel in S.
> 
> THEOREM. The inverse of a Borel bijection is a Borel bijection.
> 
> THEOREM. Let F:S into T be one-one Borel. The range of F is Borel.
> There is a Borel G:T into S such that G(F(x) = x.
> 
> DEFINITION. Partial F:S into T is Borel if and only if the inverse
> image of any Borel set in T is a Borel set in S.
> 
> THEOREM. Let F:S into T be partial. The following are equivalent.
> i. F is Borel.
> ii. dom(F) is Borel in S and graph(F) is Borel in S x T.
> iii. F is the restriction of a Borel G:S into T to a Borel set in S.
> 
> THEOREM. The inverse of every partial one-one Borel function from S
> into T is a partial one-one Borel function from T into S.
> 
> See, e.g., Kechris, Classical Descriptive Set Theory,
> 
> ************************************************************
> My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
> https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
> This is the 602nd in a series of self contained numbered
> postings to FOM covering a wide range of topics in f.o.m. The list of
> previous numbered postings #1-599 can be found at the FOM posting
> http://www.cs.nyu.edu/pipermail/fom/2015-August/018887.html
> 
> 600: Removing Deep Pathology 1  8/15/15  10:37PM
> 601: Finite Emulation Theory 1/perfect?
> 
> Harvey Friedman
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