FOM: Borel sets and Methodenreinheit (correction)
Stephen G Simpson
simpson at math.psu.edu
Sun Dec 14 19:49:53 EST 1997
In my previous posting on this, I wrote:
> Background on ATR_0:
> ...
> This ATR axiom is known to be equivalent to many ordinary
> mathematical theorems, over a weak base theory. In a sense ATR_0
> is impredicative, because any omega-model of ATR_0 necessarily
> contains sets of integers (i.e. "reals") which are not
> hyperarithmetical. In another sense ATR_0 is predicative, because
> the hyperarithmetical sets of integers form the intersection of all
> omega-models of ATR_0.
> ...
Let me revise this as follows:
Background on ATR_0:
...
The ATR axiom is known to be equivalent to many ordinary
mathematical theorems, over a weak base theory. The ATR axiom
appears to be needed in order to appropriately define and deal with
the concept of Borel set, in terms of Borel codes. More
methodological purity.
In a sense ATR_0 is impredicative, because any omega-model of ATR_0
necessarily contains sets of integers (i.e. "reals") which are not
hyperarithmetical. In another sense ATR_0 is predicative, because
such sets need not be definable in omega-models of ATR_0.
...
-- Steve
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