zbMATH — the first resource for mathematics

Projectively well-ordered inner models. (English) Zbl 0821.03023
Summary: We show that the reals in the minimal iterable inner model having \(n\) Woodin cardinals are precisely those which are \(\Delta^ 1_{n+2}\) definable from some countable ordinal. (One direction here is due to Hugh Woodin.) It follows that this model satisfies “There is a \(\Delta^ 1_{n+2}\) well-order of the reals”. We also describe some other connections between the descriptive set theory of projective sets and inner models with finitely many Woodin cardinals.

03E15 Descriptive set theory
03E60 Determinacy principles
03E55 Large cardinals
Full Text: DOI
[1] Kechris, A.S., The theory of countable analytical sets, Trans. AMS, 202, 259-297, (1975) · Zbl 0317.02082
[2] Kechris, A.S., Measure and category in effective descriptive set theory, Ann. math. logic, 5, 337-384, (1973) · Zbl 0277.02019
[3] Kechris, A.S.; Martin, D.A.; Solovay, R.M., Introduction to Q-theory, (), 199-282
[4] Martin, D.A.; Solovay, R.M., A basis theorem for σ31 sets of reals, Ann. math., 89, 138-159, (1969) · Zbl 0176.27603
[5] Martin, D.A.; Steel, J.R., A proof of projective determinacy, J. amer. math. soc., 2, 71-125, (1989) · Zbl 0668.03021
[6] D.A. Martin, Iteration trees, J. Amer. Math. Soc., to appear. · Zbl 0808.03035
[7] Mitchell, W.J.; Steel, J.R., Fine structure and iteration trees, () · Zbl 0805.03042
[8] Steel, J.R., Inner models with many Woodin cardinals, Ann. pure appl. logic, 65, 185-209, (1993) · Zbl 0805.03043
[9] Moschovakis, Y.N., Descriptive set theory, (1980), North-Holland Amsterdam · Zbl 0433.03025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.