# Difference between revisions of "Standard systems and the Scott set problem"

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Jim Schmerl <cite> schmerl1973:peano </cite> gave positive answer for arithmetic closures of sets of (arithmetic) Cohen generics of any cardinality. | Jim Schmerl <cite> schmerl1973:peano </cite> gave positive answer for arithmetic closures of sets of (arithmetic) Cohen generics of any cardinality. | ||

− | For more recent attempts employing forcing axioms see <cite> gitman2008:scott </cite> | + | For more recent attempts employing forcing axioms see <cite> engstrom2008:anote </cite> and <cite> gitman2008:scott </cite>. |

## Latest revision as of 08:11, 1 February 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Knight and Nadel [2], and independently Ehrenfeucht (unpublished), gave positive answer for $\mathfrak X$ of cardinality $\aleph_1$.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] and [5].

## 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. 117--121, Providence, R.I., 1962. MR bibtex - Julia Knight and Mark Nadel.
*Models of arithmetic and closed ideals.*J. Symbolic Logic 47(4):833--840 (1983), 1982. www DOI MR bibtex - James H. Schmerl.
*Peano models with many generic classes.*Pacific J. Math. 46:523--536, 1973. MR bibtex - Fredrik Engström.
*A note on standard systems and ultrafilters.*J. Symbolic Logic 73(3):824--830, 2008. www DOI MR bibtex - Victoria Gitman.
*Scott's problem for proper Scott sets.*J. Symbolic Logic 73(3):845--860, 2008. www DOI MR bibtex