[FOM] 171:Coordinate Free Borel Statements

[Harvey Friedman]

Here is a coordinate free version of the new Borel statements 
announced in posting #170.

THEOREM 1.(Coordinate free). Every Borel set in the plane contains or 
is disjoint from the image of an injection from the line that is 
continuous at all but countably many points.

Here is a weaker statement.

THEOREM 2.  Every Borel set in the plane contains or is disjoint from 
the image of a continuous injection from almost all real numbers into 
the plane. "Almost all" can be replaced by "a comeager set of".

Note that in Theorem 2, we only require that the injection be defined 
on a subset of the real numbers.

We cannot eliminate all points of discontinuity:

THEOREM 3. There is a Borel set in the plane such that neither it nor 
its complement contains an arc. We can replace "arc" by 
"nondegenerate path".

The statements in #170 are of course far from coordinate free. The 
following are stronger versions that I view as more technical.

THEROEM 4. Every Borel set in the plane contains or is disjoint from 
a closed set whose first or second projection contains all reals.

THEOREM 5. Every Borel set in the plane either contains a closed set 
whose first projection contains all reals, or is disjoint from a 
closed set whose second projection contains all reals.

THEOREM 6. It is necessary and sufficient to use uncountably many 
iterations of the power set operation to prove Theorem 1. The same 
holds for both forms of Theorem 2, and Theorems 4,5. All of these are 
provably equivalent in ATR0 to the existence of countable well 
founded models of the cumulative hierarchy of every countable ordinal 
length, containing any given subset of omega.

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This is the 171st 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 #1-149 can be found at 
http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html  in the FOM 
archives, 5/8/03 8:46AM. Previous ones counting from #150 are:

150:Finite obstruction/statistics  8:55AM  6/1/02
151:Finite forms by bounding  4:35AM  6/5/02
152:sin  10:35PM  6/8/02
153:Large cardinals as general algebra  1:21PM  6/17/02
154:Orderings on theories  5:28AM  6/25/02
155:A way out  8/13/02  6:56PM
156:Societies  8/13/02  6:56PM
157:Finite Societies  8/13/02  6:56PM
158:Sentential Reflection  3/31/03  12:17AM
159.Elemental Sentential Reflection  3/31/03  12:17AM
160.Similar Subclasses  3/31/03  12:17AM
161:Restrictions and Extensions  3/31/03  12:18AM
162:Two Quantifier Blocks  3/31/03  12:28PM
163:Ouch!  4/20/03  3:08AM
164:Foundations with (almost) no axioms, 4/22/0  5:31PM
165:Incompleteness Reformulated  4/29/03  1:42PM
166:Clean Godel Incompleteness  5/6/03  11:06AM
167:Incompleteness Reformulated/More  5/6/03  11:57AM
168:Incompleteness Reformulated/Again 5/8/03  12:30PM
169:New PA Independence  5:11PM  8:35PM
170:New Borel Independence  5/18/03  11:53PM