Series: Penn State Logic Seminar

Date: Tuesday, April 22, 2003

Time: 2:30 - 3:45 PM

Place: to be announced

Speaker: Chris Ciesielski

Title: 

Covering Property Axiom CPA, a Combinatorial Core of the Iterated
Perfect Set Model

Abstract: 

Many interesting mathematical properties, especially concerning real
analysis, are known to be true in the iterated perfect set (Sacks)
model, while they are false under the continuum hypothesis.  However,
the usual proofs that these facts are indeed true in this model are
very technical and involve heavy forcing machinery.  The work that I
will report in my presentation is changing this state of things.  In
the talk I will formulate a combinatorial axiom principle CPA, that is
true in the model, and demonstrate how to use it.  In particular, I
will show how CPA implies the following statements.  

1. For every subset $S$ of ${\bf R}$ of cardinality $2^\omega$ there
exists a (uniformly) continuous function $f:{\bf R}\to[0,1]$ such that
$f[S]=[0,1]$.  

2. Every perfectly meager set $S\subset{\bf R}$ has cardinality less
than~$2^\omega$.

3. Every universally null set $S\subset{\bf R}$ has cardinality less
than~$2^\omega$.

4. The cofinality of the measure ideal is less than $2^\omega$.

5.  There exists a family ${\cal F}\subset[\omega]^\omega$ of
cardinality less than $2^\omega$ which is maximal almost disjoint,
MAD. 

6. The plane ${\bf R}^2$ can be covered by less than $2^\omega$ many
sets each of which is a graph of a continuously differentiable
function from ${\bf R}$ into ${\bf R}$ in either horizontal or
vertical axis.

7. There exists a family $\cal H$ of less than $2^\omega$ pairwise
disjoint perfect sets such that $\bigcup{\cal H}$ is a linear basis of
${\bf R}$ over ${\bf Q}$.

8. There exists a non-principal selective ultrafilter on $\omega$.

The axiom will be presented in stages, starting from the simplest
form, which implies already first four of these statements.