We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in ...4, and the main conjecture of the current paper is a revision of the UB-Covering Conjecture of 4.
We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate ...the number of equivalence classes of equivalence relations obtained by countable intersections of projective sets in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.
Hod up to ADR+Θ is measurable Atmai, Rachid; Sargsyan, Grigor
Annals of pure and applied logic,
January 2019, Letnik:
170, Številka:
1
Journal Article
Recenzirano
Suppose M is a transitive class size model of ADR+“Θ is regular”. M is a minimal model of ADR+“Θ is measurable” if (i) R,Ord⊆M (ii) there is μ∈M such that M⊨“μ is a normal R-complete measure on Θ” ...and (iii) for any transitive class size N⊊M such that R⊆N, N⊨“there is no R-complete measure on Θ”. Continuing Trang's work in 8, we compute HOD of a minimal model of ADR+“Θ is measurable”.
Normal measures on large cardinals Apter, Arthur; Cummings, James
Transactions of the American Mathematical Society. Series B,
February 3, 2023, 2023-2-3, Letnik:
10, Številka:
4
Journal Article
Recenzirano
Odprti dostop
The space of normal measures on a measurable cardinal is naturally ordered by the Mitchell ordering. In the first part of this paper we show that the Mitchell ordering can be linear on a strong ...cardinal where the Generalised Continuum Hypothesis fails. In the second part we show that a supercompact cardinal at which the Generalised Continuum Hypothesis fails may carry a very large number of normal measures of Mitchell order zero.
LARGE CARDINALS BEYOND CHOICE BAGARIA, JOAN; KOELLNER, PETER; WOODIN, W. HUGH
The bulletin of symbolic logic,
09/2019, Letnik:
25, Številka:
3
Journal Article
Recenzirano
The HOD Dichotomy Theorem states that if there is an extendible cardinal, 𝛿, then either HOD is "close" to 𝑉 (in the sense that it correctly computes successors of singular cardinals greater than ...𝛿) or HOD is "far" from 𝑉 (in the sense that all regular cardinals greater than or equal to 𝛿 are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD "close" to 𝑉, or "far" from 𝑉? There is a program aimed at establishing the first alternative—the "close" side of the HOD Dichotomy. This is the program of inner model theory. In recent years the third author has provided evidence that there is an ultimate inner model—Ultimate−𝐿—and he has isolated a natural conjecture associated with the model—the Ultimate-𝐿 Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is "close" to 𝑉. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the "far" side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-𝐿 Conjecture must fail. This is the future where chaos prevails.
Long games and σ-projective sets Aguilera, Juan P.; Müller, Sandra; Schlicht, Philipp
Annals of pure and applied logic,
April 2021, 2021-04-00, Letnik:
172, Številka:
4
Journal Article
Recenzirano
Odprti dostop
We prove a number of results on the determinacy of σ-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable ...unions, and projections. We first prove the equivalence between σ-projective determinacy and the determinacy of certain classes of games of variable length <ω2 (Theorem 2.4). We then give an elementary proof of the determinacy of σ-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of the determinacy of σ-projective games of a given countable length and of games with payoff in the smallest σ-algebra containing the projective sets, from corresponding assumptions (Theorems 5.1 and 5.4).
L(R,μ) is unique Rodríguez, Daniel; Trang, Nam
Advances in mathematics (New York. 1965),
01/2018, Letnik:
324
Journal Article
Recenzirano
Odprti dostop
Under various appropriate hypotheses it is shown that there is only one determinacy model of the form L(R,μ) in which μ is a supercompact measure on Pω1(R). In particular, this gives a positive ...answer to a question asked by W.H. Woodin in 1983.
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