[FOM] 602: Removing Deep Pathology 2

[Harvey Friedman]

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