Difference between revisions of "Standard systems and the Scott set problem"
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Revision as of 12:32, 23 January 2013
The Scott set problem
A Scott set is a set of sets of natural numbers $\mathfrak X$ such that $(\omega, {\mathfrak X})\models {\sf WKL}_0$, Assume $\lnot{\sf CH}$. Is every Scott set the standard system of a nonstandard model of PA?
The origins of the problem goes back the paper of Scott [1], where it is shown that the answer is positive for countable sets $\mathfrak X$.
Knight and Nadel [2], and independently Ehrenfeucht (unpublished), gave positive answer for $\mathfrak X$ of cardinality $\aleph_1$.
Jim Schmerl [3] gave positive answer for arithmetic closures of sets of (arithmetic) Cohen generics of any cardinality.
For more recent attempts employing forcing axioms see [4]
Kanovei's question
Is there a Borel model $M\models PA$ whose standard system is the power set of $\omega$?
Woodin's question
If $\mathfrak X$ is a Borel Scott set, is there a Borel model $M\models PA$ whose standard system is $\mathfrak X$?
References
 Dana Scott. Algebras of sets binumerable in complete extensions of arithmetic. Proc. Sympos. Pure Math., Vol. V, pp. 117121, Providence, R.I., 1962. MR bibtex
 Julia Knight and Mark Nadel. Models of arithmetic and closed ideals. J. Symbolic Logic 47(4):833840 (1983), 1982. www DOI MR bibtex
 James H. Schmerl. Peano models with many generic classes. Pacific J. Math. 46:523536, 1973. MR bibtex
 Victoria Gitman. Scott's problem for proper Scott sets. J. Symbolic Logic 73(3):845860, 2008. www DOI MR bibtex