[FOM] 170:New Borel Independence

[Harvey Friedman]

We consider subsets E of the Euclidean plane, R^2.

It is standard to define the field of any set of ordered pairs as the 
set of all coordinates of its elements.

THEOREM 1. Every Borel set in the plane contains or is disjoint from 
a closed set in the plane whose field is the real line.

Here are some variants.

THEOREM 2. Every Borel set in the plane whose field is the real line 
contains or is disjoint from a closed set in the plane whose field is 
the real line.

The nice thing about Theorem 2 is that it has the following form:

*every reasonably nice set contains or is disjoint from a much nicer set*.

For instance, the following case is interesting:

*every Borel set of reals contains or is disjoint from a perfect set*.

This behaves differently, at least recursion theoretically, from the related

*every uncountable Borel set of reals contains a perfect subset*.

Presumably, one can pursue this theme systematically.

Instead of the plane, we can use any Polish space.

THEOREM 3. Every Borel set in any Polish space cross itself (whose 
field is the Polish space) contains or is disjoint from a closed set 
in the Polish space cross itself whose field is the Polish space.

We can go further, and consider products of two Polish spaces.

THEOREM 4. Every Borel set in the product of any two Polish spaces 
contains or is disjoint from a closed set in the product whose field 
contains at least one of the two spaces.

It is of some interest to specialize to the standard compact spaces, 
the closed unit interval and the Cantor set K (it doesn't make any 
difference which Cantor set is used).

THEOREM 5. Every Borel set in the unit square (whose field is the 
closed unit interval) contains or is disjoint from a compact set in 
the unit square whose field is the closed unit interval.

THEOREM 6. Every Borel set in the K x K (whose field is K) contains 
or is disjoint from a compact set in K x K whose field is K.

Finally, we state the weakest formulation.

THEOREM 7. There is an uncountable Polish space such that the 
following holds. Every Borel set in the Polish space whose field is 
the Polish space contains or is disjoint from a closed set in the 
Polish space whose field is the Polish space.

THEOREM 8. It is necessary and sufficient to use uncountably many 
iterations of the power set operation to prove Theorem 1. The same 
holds for all displayed forms of Theorems 2-6. Over ATR0, Theorems 
1-6 (all forms) are each provably equivalent 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 169th 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