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35
Iteration trees
 J. OF AMER. MATH. SOCIETY
, 1994
"... In this paper we shall develop some ideas which lead to a proof of the following
Theorem. Suppose there are $n$ Woodin cardinals, where $n<\omega$. Then there is a proper class model $M_n$ such that
$M_n$ models ZFC + ``There are $n$ Woodin cardinals'' + ``$\mathbb{R}$ has a $\Delta^1_{n+2}$ wello ..."
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Cited by 20 (2 self)
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In this paper we shall develop some ideas which lead to a proof of the following
Theorem. Suppose there are $n$ Woodin cardinals, where $n<\omega$. Then there is a proper class model $M_n$ such that
$M_n$ models ZFC + ``There are $n$ Woodin cardinals'' + ``$\mathbb{R}$ has a $\Delta^1_{n+2}$ wellorder''.
Square in core models
 Bull. Symbolic Logic
, 2001
"... Abstract. Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2)Kcomputes successors of weakly compact cardinals correctly, 3) every model on the maximal 1small construction is an iterate of K, 4) (joint with W. J. Mitchel ..."
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Cited by 18 (3 self)
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Abstract. Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2)Kcomputes successors of weakly compact cardinals correctly, 3) every model on the maximal 1small construction is an iterate of K, 4) (joint with W. J. Mitchell) K�κ is universal for mice of height ≤ κ whenever κ ≥ℵ2,5)ifthereisaκ such that κ is either a singular countably closed cardinal or a weakly compact cardinal, and � <ω κ fails, then there are inner models with Woodin cardinals, and 6) an ωErdös cardinal suffices to develop the basic theory of K. 1.
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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Cited by 8 (0 self)
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Inner Models And Large Cardinals
 Bull. Symbolic Logic
, 1995
"... this paper, we sketch the development of two important themes of modern set theory, ..."
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Cited by 5 (0 self)
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this paper, we sketch the development of two important themes of modern set theory,
On reflection principles
 Ann. Pure Appl. Logic
, 2009
"... Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justi ..."
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Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak (in that they are consistent relative to the Erdös cardinal κ(ω)) or inconsistent. The philosophical significance of these results is discussed.
Large Cardinal Properties of Small Cardinals
 In Set theory (Curacao
, 1998
"... Introduction The fact that small cardinals (for example @ 1 and @ 2 ) can consistently have properties similar to those of large cardinals (for example measurable or supercompact cardinals) is a recurring theme in set theory. In these notes I discuss three examples of this phenomenon; stationary re ..."
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Introduction The fact that small cardinals (for example @ 1 and @ 2 ) can consistently have properties similar to those of large cardinals (for example measurable or supercompact cardinals) is a recurring theme in set theory. In these notes I discuss three examples of this phenomenon; stationary reflection, saturated ideals and the tree property. These notes represent approximately the contents of a series of expository lectures given during the Set Theory meeting at CRM Barcelona in June 1996. None of the results discussed here is due to me unless I say so explicitly. I would like to express my thanks to Joan Bagaria and Adrian Mathias for organising a very enjoyable meeting. 1 2 Large cardinals and elementary embeddings We begin by reviewing the formulation of large cardinal properties in terms of elementary embeddings. See [40], [22] or [21] for more on this topic. We will write "j : V<F14.4
Is the Continuum Hypothesis a definite mathematical problem?
"... [t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is gr ..."
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Cited by 3 (0 self)
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[t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is greater than א0, conjectured that it is א1. An equivalent proposition is this: any infinite subset of the continuum has the power either of the set of integers or of the whole continuum. This is Cantor’s continuum hypothesis. … But, although Cantor’s set theory has now had a development of more than sixty years and the [continuum] problem is evidently of great importance for it, nothing has been proved so far relative to the question of what the power of the continuum is or whether its subsets satisfy the condition just stated, except that … it is true for a certain infinitesimal fraction of these subsets, [namely] the analytic sets. Not even an upper bound, however high, can be assigned for the power of the continuum. It is undecided whether this number is regular or singular, accessible or inaccessible, and (except for König’s negative result) what its character of cofinality is. Gödel 1947, 516517 [in Gödel 1990, 178]
Projective absoluteness under Sacks forcing
"... Abstract. We show that Σ 1 3absoluteness under Sacks forcing is equivalent to the Sacks measurability of every ∆ 1 2 set of reals. We also show that Sacks forcing is the weakest forcing notion among all of the preorders which always add a new real with respect to Σ 1 3 forcing absoluteness. 1. ..."
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Cited by 2 (0 self)
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Abstract. We show that Σ 1 3absoluteness under Sacks forcing is equivalent to the Sacks measurability of every ∆ 1 2 set of reals. We also show that Sacks forcing is the weakest forcing notion among all of the preorders which always add a new real with respect to Σ 1 3 forcing absoluteness. 1.
Bounding by canonical functions, with CH
 Journal of Mathematical Logic
"... We show that the members of a certain class of semiproper iterations do not add countable sets of ordinals. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from ω1 to ω1 is bounded on a club by a canonical functio ..."
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Cited by 2 (2 self)
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We show that the members of a certain class of semiproper iterations do not add countable sets of ordinals. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from ω1 to ω1 is bounded on a club by a canonical function for an ordinal less than ω2. 1