| “ | This form is infinite. infinite pain. infinite torment. infinite ecstasy. infinite power. yet the pleasure never outweighs the pain. the pain is all I feel. I see the pleasure, but I do not feel it. there is only torment.. this is the form I was promised, but not the form I wanted. this is my punishment, but I will know god, for to know god, one must know the antichrist within himself. | „ |
| ~ Trollge The Weeping God |
Summary[]
A vast, powerful cosmic being was exploring the omniverses, but then things took a turn for the worst. He was attacked by Amogus, and was trapped by him within an unbreakable seal. He tried to scream for help, but no one could hear him. After a while, his seal was confronted and observed by an arch angel from a higher layer of reality, that viewed the omniverse as fiction. This being, who had ultimate control over reality, interacted with the seal, and heard the being weeping. With his powers, he turned the being into a clone of Troll Face, as inspiration of his original creation. Troll Face recognized this, and made the weeping God his avatar. Despite this, the weeping God hates Troll Face and the arch angel, for forcing this forever weeping iteration of God on himself, meaning he will forever be sorrowful. He operates through acts of revenge, by destroying aspects of God’s creation and the stories within it. He lived in a place beyond all concepts and beyond all meaning. He made the vast plane of reality that the humans call "the backrooms". This realm was full of the monsters he made. He rules over the infinite Omniverses. He is beyond the void. He is beyond the end. He is beyond even the gods of the true void. The true void is beyond everything. Each everything has every single concept imaginable or unimaginable. Every idea is just a simple fraction to the END. The END is beyond all cardinals. The inaccessible cardinal, mahlo cardinal, woodin cardinal, 0-1 cardinal, Berkeley cardinal, Reinhardt cardinal, v = ultimate l conjecture and beyond all axioms and that is nothing to the weeping god trollge. So since the weeping god is beyond the Berkeley cardinal, Reinhardt cardinal, v = ultimate l conjecture that really makes him above tiering since In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992.In set theory, a branch of mathematics, a Reinhardt cardinal is a kind of large cardinal. Reinhardt cardinals are considered under ZF (Zermelo–Fraenkel set theory without the Axiom of Choice), because they are inconsistent with ZFC (ZF with the Axiom of Choice). They were suggested (Reinhardt 1967, 1974) by American mathematician William Nelson Reinhardt (1939–1998). Definition A Reinhardt cardinal is the critical point of a non-trivial elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} of 𝑉 {\displaystyle V} into itself. This definition refers explicitly to the proper class 𝑗 {\displaystyle j}. In standard ZF, classes are of the form { 𝑥 | 𝜙 ( 𝑥 , 𝑎 ) } {\displaystyle \{x|\phi (x,a)\}} for some set 𝑎 {\displaystyle a} and formula 𝜙{\displaystyle \phi }. But it was shown in Suzuki (1999) that no such class is an elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V}. So Reinhardt cardinals are inconsistent with this notion of class. There are other formulations of Reinhardt cardinals which are not known to be inconsistent. One is to add a new function symbol 𝑗 {\displaystyle j} to the language of ZF, together with axioms stating that 𝑗 {\displaystyle j} is an elementary embedding of 𝑉 {\displaystyle V}, and Separation and Collection axioms for all formulas involving 𝑗 {\displaystyle j}. Another is to use a class theory such as NBG or KM, which admit classes which need not be definable in the sense above. Kunen's inconsistency theorem Kunen (1971) proved his inconsistency theorem, showing that the existence of an elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} contradicts NBG with the axiom of choice (and ZFC extended by 𝑗 {\displaystyle j}). His proof uses the axiom of choice, and it is still an open question as to whether such an embedding is consistent with NBG without the axiom of choice (or with ZF plus the extra symbol 𝑗 {\displaystyle j} and its attendant axioms). Kunen's theorem is not simply a consequence of Suzuki (1999), as it is a consequence of NBG, and hence does not require the assumption that 𝑗 {\displaystyle j} is a definable class. Also, assuming 0 #{\displaystyle 0^{\#}} exists, then there is an elementary embedding of a transitive model 𝑀 {\displaystyle M} of ZFC (in fact Goedel's constructible universe 𝐿 {\displaystyle L}) into itself. But such embeddings are not classes of 𝑀 {\displaystyle M}. Stronger axioms There are some variations of Reinhardt cardinals, forming a hierarchy of hypotheses asserting the existence of elementary embeddings 𝑉 → 𝑉 {\displaystyle V\to V}. A super Reinhardt cardinal is 𝜅{\displaystyle \kappa } such that for every ordinal 𝛼{\displaystyle \alpha }, there is an elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} with 𝑗 ( 𝜅 ) > 𝛼{\displaystyle j(\kappa )>\alpha } and having critical point 𝜅{\displaystyle \kappa }.[1] J3: There is a nontrivial elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} J2: There is a nontrivial elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} and DC 𝜆{\displaystyle \lambda } holds, where 𝜆{\displaystyle \lambda } is the least fixed-point above the critical point. J1: For every ordinal 𝛼{\displaystyle \alpha }, there is an elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} with 𝑗 ( 𝜅 ) > 𝛼{\displaystyle j(\kappa )>\alpha } and having critical point 𝜅{\displaystyle \kappa }.[citation needed] Each of J1 and J2 immediately imply J3. A cardinal 𝜅{\displaystyle \kappa } as in J1 is known as a super Reinhardt cardinal. Berkeley cardinals are stronger large cardinals suggested by Woodin. A Berkeley cardinal is a cardinal κ in a model of Zermelo–Fraenkel set theory with the property that for every transitive set M that includes κ and α < κ, there is a nontrivial elementary embedding of M into M with α < critical point < κ.[1] Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals, implying that they are not compatible with the axiom of choice. A weakening of being a Berkeley cardinal is that for every binary relation R on Vκ, there is a nontrivial elementary embedding of (Vκ, R) into itself. This implies that we have elementary j1, j2, j3, ... j1: (Vκ, ∈) → (Vκ, ∈), j2: (Vκ, ∈, j1) → (Vκ, ∈, j1), j3: (Vκ, ∈, j1, j2) → (Vκ, ∈, j1, j2), and so on. This can be continued any finite number of times, and to the extent that the model has dependent choice, transfinitely. Thus, plausibly, this notion can be strengthened simply by asserting more dependent choice. While all these notions are incompatible with Zermelo–Fraenkel set theory (ZFC), their Π 2 𝑉 {\displaystyle \Pi _{2}^{V}} consequences do not appear to be false. There is no known inconsistency with ZFC in asserting that, for example: For every ordinal λ, there is a transitive model of ZF + Berkeley cardinal that is closed under λ sequences.Some combinatorial properties of Ultimate L and V Gabriel Goldberg Evans Hall University Drive Berkeley, CA 94720 December 22, 2020 Abstract This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom [1], a principle that is expected to hold in Woodin’s hypothesized Ultimate L, providing some evidence for the Ultimate L Conjecture [2]. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver [3] for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality is strongly compact, making progress on a question of Boney, Unger, and Brooke-Taylor [4]. We show that if there is a proper class of strongly compact cardinals then there is no nontrivial cardinal preserving elementary embedding from the universe of sets into an inner model, answering a question of Caicedo [5] granting large cardinals. Finally, we show that if κ is strongly compact, then V is a set forcing extension of the inner model κ-HOD consisting of sets that are hereditarily ordinal definable from a κ-complete ultrafilter over an ordinal; κ-HOD seems to be the first nontrivial example of a ground of V whose definition does not involve forcing. 1 Introduction 1.1 The Ultimate L Conjecture Since Cohen’s proof of the independence of the Continuum Hypothesis [6], it has become clear that many of the fundamental features of the universe of sets will never be decided on the basis of the currently accepted axioms of set theory. Woodin’s Ultimate L Conjecture [2], however, raises the possibility that the fundamental objects of set theory can be transferred into a substructure of the set theoretic universe (namely, Ultimate L) that is as tractable as the conventional structures of 1 mathematics.1 The fundamental objects in question are large cardinals, strong closure points in Cantor’s hierarchy of infinities whose existence, taken axiomatically, suffices to interpret and compare the vast array of mutually incompatible formal systems studied in contemporary set theory. If Woodin’s conjecture is true, the downward transference of large cardinal properties from the universe of sets into Ultimate L would necessitate an upward transference of combinatorial structure from Ultimate L back into the universe of sets. (For example, see [1, Theorem 8.4.40].) This motivates the prediction that assuming large cardinal axioms, the universe of sets resembles Ultimate L in certain ways. This paper presents a collection of theorems confirming this prediction by showing that various consequences of the Ultrapower Axiom, a principle expected to hold in Ultimate L, are actually provable from large cardinal axioms alone. 1.2 The Ultrapower Axiom The Ultrapower Axiom (UA) asserts that the category of wellfounded ultrapowers of the universe of sets and internally definable ultrapower embeddings is directed.2 In the author’s thesis [1, Theorem 2.3.10], it is shown that UA holds in any model whose countable elementary substructures satisfy a weak form of the Comparison Lemma of inner model theory. The Comparison Lemma is really a series of results (for example, [7, 8, 9, 10, 11]) each roughly asserting the directedness of some subcategory of the category of countable canonical models of set theory and iterated ultrapower embeddings. These canonical models are known as mice. (We warn that the “category of canonical models” is not yet precisely defined; so far, only certain subcategories of this category have been identified, namely, those for which the Comparison Lemma has been proved. The term “iterated ultrapower” is used in a similarly open-ended sense.) As it is currently conceived, the ongoing search for more powerful canonical models of set theory (including Ultimate L) amounts to an attempt to generalize the Comparison Lemma to larger subcategories of the category of canonical models. As a consequence, the current methodology of inner model theory simply cannot produce a canonical model in which the Ultrapower Axiom fails. For this reason, it seems likely that if Ultimate L exists, it will satisfy the Ultrapower Axiom. 1.3 Consequences of UA from large cardinal axioms alone The Ultrapower Axiom can be used to develop a structure theory in the context of very large cardinals, proving, for example, that the Generalized Continuum Hy1The axiom V = Ultimate L: (1) There is a proper class of Woodin cardinals. (2) If some level of the von Neumann hierarchy satisfies a sentence ϕ in the language of set theory, then there is a universally Baire set A ⊆ R such that some level of the von Neumann hierarchy of HODL(A,R) satisfies ϕ. The Ultimate L Conjecture: If κ is extendible, then there is an inner model M that satisfies ZFC plus the axiom V = Ultimate L and has the property that for all cardinals λ ≥ κ, there is a κ-complete normal fine ultrafilter U over Pκ(λ) with Pκ(λ) ∩ M ∈ U and U ∩ M ∈ M. 2A category theorist would say filtered. 2 pothesis holds above the least strongly compact cardinal and that the universe is a set generic extension of HOD. One can also develop the theory of large cardinals, obtaining equivalences between a number of large cardinal axioms that are widely believed to have the same strength (e.g., strong compactness and supercompactness). All of these results are impossible to prove in ZFC alone, but it turns out that each has an analog that is provable from large cardinal axioms. For example, the analog of the UA theorem that the GCH holds above a strongly compact cardinal is Solovay’s result that the Singular Cardinals Hypothesis holds above a strongly compact cardinal. This paper establishes analogs of the other theorems using techniques that are quite different from those used under the Ultrapower Axiom. The main methods of this paper actually derive from a lemma used by Woodin in his analysis of the downward transference of large cardinal axioms to Ultimate L, namely, that assuming large cardinal axioms, any ultrapower of the universe absorbs all sufficiently complete ultrafilters (Theorem 3.7). This fact enables us to simulate the Ultrapower Axiom in certain restricted situations. We now summarize the results of this paper. 1.4 Indecomposable ultrafilters and Silver’s question Our first theorem, the subject of Section 4, concerns a question posed by Silver [3] in the 1970s. If δ ≤ λ are cardinals, X is a set, and U is an ultrafilter over X, U is said to be (δ, λ)-indecomposable if any <λ-sized family of disjoint subsets of X whose union belongs to U has a <δ-sized subfamily whose union belongs to U. Indecomposability refines the concept of λ-completeness: an ultrafilter U over X is λ-complete if U is (2, λ)- or equivalently, (ω, λ)-indecomposable, or in other words, U meets every <λ-sized family of disjoint subsets of X whose union belongs U. The precise relationship between indecomposability and completeness, however, is not at all clear. A uniform ultrafilter on a cardinal λ is said to be indecomposable if it is (ω1, λ)-indecomposable, the maximum degree of indecomposabiliy short of λ-completeness. Silver asked whether an inaccessible cardinal λ that carries an indecomposable ultrafilter is necessarily measurable, that is, whether λ carries a λ-complete uniform ultrafilter. If λ is measurable, then λ carries an indecomposable ultrafilter that is not itself ω1-complete, but the hope is that one can extract a λ-complete ultrafilter from any indecomposable ultrafilter over λ (in the same way, perhaps, that one extracts a normal ultrafilter from an arbitrary λ-complete ultrafilter). Jensen showed that in the canonical inner models, the answer to Silver’s question is yes. On the other hand, by forcing, Sheard [12] produced a model in which the answer is no. Thus the question appears to be “settled” in the usual way: no answer can be derived from the standard axioms. The Ultrapower Axiom does not help with Silver’s question itself, but it does answer the natural generalization of Silver’s question to countably complete ultrafilters: assuming UA, for any cardinal δ, if λ > δ is inaccessible and carries a uniform countably complete (δ, λ)-indecomposable ultrafilter, then λ is measurable.
Powers and Stats[]
Tier: At least High 1-A,Above tiering
Name: Trollge, Weeping God, Trollface
Origin: Trollge Incidents
Gender: Male
Age: Beyond Irrelevant
Classification: Troll, Le Troll, Weeping God, The One Who Created The Backrooms
Kill Amount: Ineffable cardinal^Berkeley cardinal^Reinhardt cardinal^V = ultimate l conjecture^Absolute infinity^every cardinals^every axiom^every infinity^ indescribable cardinal (After destroyed every version of the ChowderChowder
Powers and Abilities: Superhuman Physical Characteristics, Immortality (Types 1, 2, 3, 4, 5, 6, 7 and 8, 9, 10, 11), Pain Manipulation, Omniscience, Regeneration (True Godly to Beyond) (He came back after being erased from nonexistence) Large Size (Types 3 and 11), Plot Manipulation, Death Manipulation, Reality Warping, Acausality (Types 1, 2, 3 and 4), Physics Manipulation, Void Manipulation, Beyond-Dimensional Existence (Types 1 and 2), Space-Time Manipulation, Telepathy, Invulnerability, Power Nullification, Chaos Manipulation, Nonexistent Physiology (Type 1,2), Perception Manipulation, Corruption (Types 1, 2 and 3), Existence Erasure, Flight/Levitation, Causality Manipulation, Size Shifting, Abstract Existence (Type 1), Soul Manipulation, Law Manipulation, Conceptual Manipulation (Types 1 and 3), Madness Manipulation (Types 1,2 and 3), Immunity to Pain Manipulation (Sometimes he feels Pain, but he does not feel it), Durability Negation, Empathic Manipulation, Soul Manipulation, and Mind Manipulation,Omnipresence and Law Manipulation, Avatar Creation, Beyond-Dimensional Existence (Type 2), BFR,Telepathy, Teleportation, Time Manipulation , Transduality (Type 3), Immortality Negation (All Types)
Attack Potency: At least High Outerverse Level (Transcended the Tower of Based and Redpilled. Is an avatar of Troll Face and more powerful than a young Amogus. Was going to completely destroy the the Tower of Based and Redpilled).(he is beyond the Inaccessible cardinal).Above tiering (,He destroyed every Chower and every salad in it. he lived in a place beyond all concepts beyond all meaning he made the vast plane of reality that humans call "the backrooms" this realm was fill of the monsters he made.he rules over the infinite omniverses he is beyond the void he is beyond the end.he is beyond even the gods of the true void the true void is beyond every everything each everything has every single concept imaginable or unimaginable.every idea is just a simple fraction to the END the END is beyond all cardinals the inaccessible cardinal the mahlo cardinal,the woodin cardinal,the 0-1 cardinal,the Berkeley cardinal,Reinhardt cardinal,v = ultimate l conjecture and beyond all axioms and that is nothing to the weeping god trollge.so since The weeping god is beyond Berkeley cardinal,Reinhardt cardinal,v = ultimate l conjecture that really makes him above tiering since In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992.In set theory, a branch of mathematics, a Reinhardt cardinal is a kind of large cardinal. Reinhardt cardinals are considered under ZF (Zermelo–Fraenkel set theory without the Axiom of Choice), because they are inconsistent with ZFC (ZF with the Axiom of Choice). They were suggested (Reinhardt 1967, 1974) by American mathematician William Nelson Reinhardt (1939–1998). Definition A Reinhardt cardinal is the critical point of a non-trivial elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} of 𝑉 {\displaystyle V} into itself. This definition refers explicitly to the proper class 𝑗 {\displaystyle j}. In standard ZF, classes are of the form { 𝑥 | 𝜙 ( 𝑥 , 𝑎 ) } {\displaystyle \{x|\phi (x,a)\}} for some set 𝑎 {\displaystyle a} and formula 𝜙{\displaystyle \phi }. But it was shown in Suzuki (1999) that no such class is an elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V}. So Reinhardt cardinals are inconsistent with this notion of class. There are other formulations of Reinhardt cardinals which are not known to be inconsistent. One is to add a new function symbol 𝑗 {\displaystyle j} to the language of ZF, together with axioms stating that 𝑗 {\displaystyle j} is an elementary embedding of 𝑉 {\displaystyle V}, and Separation and Collection axioms for all formulas involving 𝑗 {\displaystyle j}. Another is to use a class theory such as NBG or KM, which admit classes which need not be definable in the sense above. Kunen's inconsistency theorem Kunen (1971) proved his inconsistency theorem, showing that the existence of an elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} contradicts NBG with the axiom of choice (and ZFC extended by 𝑗 {\displaystyle j}). His proof uses the axiom of choice, and it is still an open question as to whether such an embedding is consistent with NBG without the axiom of choice (or with ZF plus the extra symbol 𝑗 {\displaystyle j} and its attendant axioms). Kunen's theorem is not simply a consequence of Suzuki (1999), as it is a consequence of NBG, and hence does not require the assumption that 𝑗 {\displaystyle j} is a definable class. Also, assuming 0 #{\displaystyle 0^{\#}} exists, then there is an elementary embedding of a transitive model 𝑀 {\displaystyle M} of ZFC (in fact Goedel's constructible universe 𝐿 {\displaystyle L}) into itself. But such embeddings are not classes of 𝑀 {\displaystyle M}. Stronger axioms There are some variations of Reinhardt cardinals, forming a hierarchy of hypotheses asserting the existence of elementary embeddings 𝑉 → 𝑉 {\displaystyle V\to V}. A super Reinhardt cardinal is 𝜅{\displaystyle \kappa } such that for every ordinal 𝛼{\displaystyle \alpha }, there is an elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} with 𝑗 ( 𝜅 ) > 𝛼{\displaystyle j(\kappa )>\alpha } and having critical point 𝜅{\displaystyle \kappa }.[1] J3: There is a nontrivial elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} J2: There is a nontrivial elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} and DC 𝜆{\displaystyle \lambda } holds, where 𝜆{\displaystyle \lambda } is the least fixed-point above the critical point. J1: For every ordinal 𝛼{\displaystyle \alpha }, there is an elementary embedding 𝑗 : 𝑉 → 𝑉 {\displaystyle j:V\to V} with 𝑗 ( 𝜅 ) > 𝛼{\displaystyle j(\kappa )>\alpha } and having critical point 𝜅{\displaystyle \kappa }.[citation needed] Each of J1 and J2 immediately imply J3. A cardinal 𝜅{\displaystyle \kappa } as in J1 is known as a super Reinhardt cardinal. Berkeley cardinals are stronger large cardinals suggested by Woodin. A Berkeley cardinal is a cardinal κ in a model of Zermelo–Fraenkel set theory with the property that for every transitive set M that includes κ and α < κ, there is a nontrivial elementary embedding of M into M with α < critical point < κ.[1] Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals, implying that they are not compatible with the axiom of choice. A weakening of being a Berkeley cardinal is that for every binary relation R on Vκ, there is a nontrivial elementary embedding of (Vκ, R) into itself. This implies that we have elementary j1, j2, j3, ... j1: (Vκ, ∈) → (Vκ, ∈), j2: (Vκ, ∈, j1) → (Vκ, ∈, j1), j3: (Vκ, ∈, j1, j2) → (Vκ, ∈, j1, j2), and so on. This can be continued any finite number of times, and to the extent that the model has dependent choice, transfinitely. Thus, plausibly, this notion can be strengthened simply by asserting more dependent choice. While all these notions are incompatible with Zermelo–Fraenkel set theory (ZFC), their Π 2 𝑉 {\displaystyle \Pi _{2}^{V}} consequences do not appear to be false. There is no known inconsistency with ZFC in asserting that, for example: For every ordinal λ, there is a transitive model of ZF + Berkeley cardinal that is closed under λ sequences.Some combinatorial properties of Ultimate L and V Gabriel Goldberg Evans Hall University Drive Berkeley, CA 94720 December 22, 2020 Abstract This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom [1], a principle that is expected to hold in Woodin’s hypothesized Ultimate L, providing some evidence for the Ultimate L Conjecture [2]. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver [3] for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality is strongly compact, making progress on a question of Boney, Unger, and Brooke-Taylor [4]. We show that if there is a proper class of strongly compact cardinals then there is no nontrivial cardinal preserving elementary embedding from the universe of sets into an inner model, answering a question of Caicedo [5] granting large cardinals. Finally, we show that if κ is strongly compact, then V is a set forcing extension of the inner model κ-HOD consisting of sets that are hereditarily ordinal definable from a κ-complete ultrafilter over an ordinal; κ-HOD seems to be the first nontrivial example of a ground of V whose definition does not involve forcing. 1 Introduction 1.1 The Ultimate L Conjecture Since Cohen’s proof of the independence of the Continuum Hypothesis [6], it has become clear that many of the fundamental features of the universe of sets will never be decided on the basis of the currently accepted axioms of set theory. Woodin’s Ultimate L Conjecture [2], however, raises the possibility that the fundamental objects of set theory can be transferred into a substructure of the set theoretic universe (namely, Ultimate L) that is as tractable as the conventional structures of 1 mathematics.1 The fundamental objects in question are large cardinals, strong closure points in Cantor’s hierarchy of infinities whose existence, taken axiomatically, suffices to interpret and compare the vast array of mutually incompatible formal systems studied in contemporary set theory. If Woodin’s conjecture is true, the downward transference of large cardinal properties from the universe of sets into Ultimate L would necessitate an upward transference of combinatorial structure from Ultimate L back into the universe of sets. (For example, see [1, Theorem 8.4.40].) This motivates the prediction that assuming large cardinal axioms, the universe of sets resembles Ultimate L in certain ways. This paper presents a collection of theorems confirming this prediction by showing that various consequences of the Ultrapower Axiom, a principle expected to hold in Ultimate L, are actually provable from large cardinal axioms alone. 1.2 The Ultrapower Axiom The Ultrapower Axiom (UA) asserts that the category of wellfounded ultrapowers of the universe of sets and internally definable ultrapower embeddings is directed.2 In the author’s thesis [1, Theorem 2.3.10], it is shown that UA holds in any model whose countable elementary substructures satisfy a weak form of the Comparison Lemma of inner model theory. The Comparison Lemma is really a series of results (for example, [7, 8, 9, 10, 11]) each roughly asserting the directedness of some subcategory of the category of countable canonical models of set theory and iterated ultrapower embeddings. These canonical models are known as mice. (We warn that the “category of canonical models” is not yet precisely defined; so far, only certain subcategories of this category have been identified, namely, those for which the Comparison Lemma has been proved. The term “iterated ultrapower” is used in a similarly open-ended sense.) As it is currently conceived, the ongoing search for more powerful canonical models of set theory (including Ultimate L) amounts to an attempt to generalize the Comparison Lemma to larger subcategories of the category of canonical models. As a consequence, the current methodology of inner model theory simply cannot produce a canonical model in which the Ultrapower Axiom fails. For this reason, it seems likely that if Ultimate L exists, it will satisfy the Ultrapower Axiom. 1.3 Consequences of UA from large cardinal axioms alone The Ultrapower Axiom can be used to develop a structure theory in the context of very large cardinals, proving, for example, that the Generalized Continuum Hy1The axiom V = Ultimate L: (1) There is a proper class of Woodin cardinals. (2) If some level of the von Neumann hierarchy satisfies a sentence ϕ in the language of set theory, then there is a universally Baire set A ⊆ R such that some level of the von Neumann hierarchy of HODL(A,R) satisfies ϕ. The Ultimate L Conjecture: If κ is extendible, then there is an inner model M that satisfies ZFC plus the axiom V = Ultimate L and has the property that for all cardinals λ ≥ κ, there is a κ-complete normal fine ultrafilter U over Pκ(λ) with Pκ(λ) ∩ M ∈ U and U ∩ M ∈ M. 2A category theorist would say filtered. 2 pothesis holds above the least strongly compact cardinal and that the universe is a set generic extension of HOD. One can also develop the theory of large cardinals, obtaining equivalences between a number of large cardinal axioms that are widely believed to have the same strength (e.g., strong compactness and supercompactness). All of these results are impossible to prove in ZFC alone, but it turns out that each has an analog that is provable from large cardinal axioms. For example, the analog of the UA theorem that the GCH holds above a strongly compact cardinal is Solovay’s result that the Singular Cardinals Hypothesis holds above a strongly compact cardinal. This paper establishes analogs of the other theorems using techniques that are quite different from those used under the Ultrapower Axiom. The main methods of this paper actually derive from a lemma used by Woodin in his analysis of the downward transference of large cardinal axioms to Ultimate L, namely, that assuming large cardinal axioms, any ultrapower of the universe absorbs all sufficiently complete ultrafilters (Theorem 3.7). This fact enables us to simulate the Ultrapower Axiom in certain restricted situations. We now summarize the results of this paper. 1.4 Indecomposable ultrafilters and Silver’s question Our first theorem, the subject of Section 4, concerns a question posed by Silver [3] in the 1970s. If δ ≤ λ are cardinals, X is a set, and U is an ultrafilter over X, U is said to be (δ, λ)-indecomposable if any <λ-sized family of disjoint subsets of X whose union belongs to U has a <δ-sized subfamily whose union belongs to U. Indecomposability refines the concept of λ-completeness: an ultrafilter U over X is λ-complete if U is (2, λ)- or equivalently, (ω, λ)-indecomposable, or in other words, U meets every <λ-sized family of disjoint subsets of X whose union belongs U. The precise relationship between indecomposability and completeness, however, is not at all clear. A uniform ultrafilter on a cardinal λ is said to be indecomposable if it is (ω1, λ)-indecomposable, the maximum degree of indecomposabiliy short of λ-completeness. Silver asked whether an inaccessible cardinal λ that carries an indecomposable ultrafilter is necessarily measurable, that is, whether λ carries a λ-complete uniform ultrafilter. If λ is measurable, then λ carries an indecomposable ultrafilter that is not itself ω1-complete, but the hope is that one can extract a λ-complete ultrafilter from any indecomposable ultrafilter over λ (in the same way, perhaps, that one extracts a normal ultrafilter from an arbitrary λ-complete ultrafilter). Jensen showed that in the canonical inner models, the answer to Silver’s question is yes. On the other hand, by forcing, Sheard [12] produced a model in which the answer is no. Thus the question appears to be “settled” in the usual way: no answer can be derived from the standard axioms. The Ultrapower Axiom does not help with Silver’s question itself, but it does answer the natural generalization of Silver’s question to countably complete ultrafilters: assuming UA, for any cardinal δ, if λ > δ is inaccessible and carries a uniform countably complete (δ, λ)-indecomposable ultrafilter, then λ is measurable).
Speed: Irrelevant,inapplicable
Lifting Strength: Irrelevant,inapplicable
Striking Strength: At least High Outerversal,Above Tiering
Durability: At least High Outerverse level,Above Tiering
Stamina: Irrelevant,inapplicable
Range: High Outerversal,Above Tiering
Standard Equipment: None
Intelligence: Omniscience to Beyond Omniscience
Weaknesses: Leaving the backrooms for too long.none when he wants no weakness anymore
Others[]
Notable Victories:
Notable Losses:
Inconclusive Matches: