A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set fo... more A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of zfc has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion v=hod that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent.
We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the grou... more We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the ground model to be the η-th inner mantle of the extension, in which the sequence of inner mantles has length at least η. This answers a conjecture of Fuchs, Hamkins, and Reitz [FHR15] in the positive. We also show that M(η) forces the ground model to be the η-th iterated HOD of the extension, where the sequence of iterated HODs has length at least η. We conclude by showing that the lengths of the sequences of inner mantles and of iterated HODs can be separated to be any two ordinals you please.
We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the grou... more We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the ground model to be the η-th inner mantle of the extension, in which the sequence of inner mantles has length at least η. This answers a conjecture of Fuchs, Hamkins, and Reitz [FHR15] in the positive. We also show that M(η) forces the ground model to be the η-th iterated HOD of the extension, where the sequence of iterated HODs has length at least η. We conclude by showing that the lengths of the sequences of inner mantles and of iterated HODs can be separated to be any two ordinals you please.
This paper explores how a pluralist view can arise in a natural way out of the day-today practice... more This paper explores how a pluralist view can arise in a natural way out of the day-today practice of modern set theory. By contrast, the widely accepted orthodox view is that there is an ultimate universe of sets V , and it is in this universe that mathematics takes place. From this view, the purpose of set theory is "learning the truth about V ." It has become apparent, however, that the phenomenon of independence-those questions left unresolved by the axioms-holds a central place in the investigation. This paper introduces the notion of independence, explores the primary tool ("soundness") for establishing independence results, and shows how a plurality of models arises through the investigation of this phenomenon. Building on a familiar example from Euclidean geometry, a template for independence proofs is established. Applying this template in the domain of set theory leads to a consideration of forcing, the tool par excellence for constructing universes of sets. Fifty years of forcing has resulted in a profusion of universes exhibiting a wide variety of characteristics-a multiverse of set theories. Direct study of this multiverse presents technical challenges due to its second-order nature. Nonetheless, there are certain nice "local neighborhoods" of the multiverse that are amenable to first-order analysis, and set-theoretic geology studies just such a neighborhood, the collection of grounds of a given universe V of set theory. I will explore some of the properties of this collection, touching on major concepts, open questions, and recent developments.
A pointwise definable model is one in which every object is definable without parameters. In a mo... more A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. 2000 Mathematics Subject Classification. 03E55. Key words and phrases. set theory, forcing. 1 See [Ani] for an instance of the argument at MathOverflow, which surely serves a brisk cup of math tea online. We leave aside the remark of Horatio, eight-year-old son of the first author, who announced, "Sure, papa, I can describe any number. Let me show you: tell me any number, and I'll tell you a description of it!"
In light of the celebrated theorem of Vopěnka [VH72], proving in ZFC that every set is generic ov... more In light of the celebrated theorem of Vopěnka [VH72], proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe V must be a class-forcing extension of HOD by some possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its HOD for any class forcing notion definable in HOD and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman [Fri12] showed, positively, that if one augments HOD with a certain ZFC-amenable class A, definable in V , then the set-theoretic universe V is a class-forcing extension of the expanded structure HOD, ∈, A. Our result shows that this augmentation process can be necessary. The same example shows that V is not necessarily a class-forcing extension of the mantle, and the method provides counterexamples to the intermediate model property, namely, a class-forcing extension V ⊆ W ⊆ V [G] with an intermediate transitive inner model W that is neither a class-forcing extension of V nor a ground model of V [G] by any definable class forcing notion with definable forcing relations.
Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a su... more Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a subset to κ by forcing: the Cohen poset Add(κ, 1), and the Cohen poset of the inner model Add(κ, 1) W. The forcing from W will be at least as strong as the forcing from V (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from V to fail to be as strong as that from W. The results are generalized to Add(κ, λ), and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.
Axiom is first-order expressible, and any model of zfc has a class-forcing extension which satisf... more Axiom is first-order expressible, and any model of zfc has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion v=hod that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing. First and foremost I would like to thank my advisor, Joel David Hamkins, an inspiration in mathematics and in life. He sets high standards for his students and conforms to even higher standards himself. An aspiring mathematician could not ask for a better mentor. Thank you also to my committee for their time, attention, and insightful comments. My close friends and respected colleagues Victoria Gitman and Thomas Johnstone, who have shared five years of intense and wonderful study with me, deserve much credit for enabling me to produce this work. Thank you for listening to my ideas and sharing your own. Learning has never been such an adventure and such a pleasure as it has been with you. Thank you to my family for instilling the crazy idea that I could accomplish whatever I wanted, and for teaching me about love. Finally, to my wife and best friend Gwen I owe the gratitude of uncomplaining support and unconditional love through far too many trials.
We introduce and consider the inner-model reflection principle, which asserts that whenever a sta... more We introduce and consider the inner-model reflection principle, which asserts that whenever a statement ϕ(a) in the first-order language of set theory is true in the set-theoretic universe V , then it is also true in a proper inner model W V. A stronger principle, the ground-model reflection principle, asserts that any such ϕ(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the innermodel hypothesis IMH. We also consider some questions concerning the expressibility of the principles.
We introduce and consider the inner-model reflection principle, which asserts that whenever a sta... more We introduce and consider the inner-model reflection principle, which asserts that whenever a statement ϕ(a) in the first-order language of set theory is true in the set-theoretic universe V , then it is also true in a proper inner model W V. A stronger principle, the ground-model reflection principle, asserts that any such ϕ(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π 2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.
A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC... more A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, wh...
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set fo... more A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent.
A pointwise definable model is one in which every object is definable without parameters. In a mo... more A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. 2000 Mathematics Subject Classification. 03E55. Key words and phrases. set theory, forcing. 1 See [Ani] for an instance of the argument at MathOverflow, which surely serves a brisk cup of math tea online. We leave aside the remark of Horatio, eight-year-old son of the first author, who announced, "Sure, papa, I can describe any number. Let me show you: tell me any number, and I'll tell you a description of it!"
A ground of the universe V is a transitive proper class W ⊆ V , such that W |= ZFC and V is obtai... more A ground of the universe V is a transitive proper class W ⊆ V , such that W |= ZFC and V is obtained by set forcing over W , so that V = W [G] for some W-generic filter G ⊆ P ∈ W. The model V satisfies the ground axiom GA if there are no such W properly contained in V. The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V. The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V. The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.
Abstract. The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of... more Abstract. The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, it is first-order expressible in set theory. The previously known models of the Ground Axiom all satisfy strong forms of V =HOD. In this article, we show that the Ground Axiom is relatively consistent with V ̸ = HOD. In fact, every model of ZFC has a class-forcing extension that is a model of ZFC + GA + V ̸ = HOD. The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a classforcing extension with ZFC + GA + V ̸ = HOD in which this supercompact cardinal is preserved. The Ground Axiom, introduced by Hamkins and Reitz [10, 9, 4], is the assertion that the universe of set theory is not a nontrivial set-forcing extension of any inner model. That is, the Ground Axiom asserts that if W is an inner model of the universe V and G is W-generic for nontrivial forcing, th...
In light of the celebrated theorem of Vop\v{e}nka (1972), proving in ZFC that every set is generi... more In light of the celebrated theorem of Vop\v{e}nka (1972), proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of HOD by some possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its HOD for any class forcing notion definable in HOD and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman (2012) showed, positively, that if one augments HOD with a certain ZFC-amenable class $A$, definable in $V$, then the set-theoretic universe $V$ is a class-forcing extension of the expanded structure $\langle\text{HOD},\in,A\rangle$. Our result shows that this augmentation process can be necessary. The same example shows that $V$ is not necessarily a class-forcing extension of the mantle, and the method provides counterexamples to the intermediate model property, n...
Axiom is first-order expressible, and any model of zfc has a class-forcing extension which satisf... more Axiom is first-order expressible, and any model of zfc has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion v=hod that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing. First and foremost I would like to thank my advisor, Joel David Hamkins, an inspiration in mathematics and in life. He sets high standards for his students and conforms to even higher standards himself. An aspiring mathematician could not ask for a better mentor. Thank you also to my committee for their time, attention, and insightful comments. My close friends and respected colleagues Victoria Gitman and Thomas Johnstone, who have shared five years of intense and wonderful study with me, deserve much credit for enabling me to produce this work. Thank you for listening to my ideas and sharing your own. Learning has never been such an adventure and such a pleasure as it has been with you. Thank you to my family for instilling the crazy idea that I could accomplish whatever I wanted, and for teaching me about love. Finally, to my wife and best friend Gwen I owe the gratitude of uncomplaining support and unconditional love through far too many trials.
Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a su... more Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a subset to κ by forcing: the Cohen poset Add(κ, 1), and the Cohen poset of the inner model Add(κ, 1) W . The forcing from W will be at least as strong as the forcing from V (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from V to fail to be as strong as that from W . The results are generalized to Add(κ, λ), and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set fo... more A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of zfc has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion v=hod that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent.
We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the grou... more We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the ground model to be the η-th inner mantle of the extension, in which the sequence of inner mantles has length at least η. This answers a conjecture of Fuchs, Hamkins, and Reitz [FHR15] in the positive. We also show that M(η) forces the ground model to be the η-th iterated HOD of the extension, where the sequence of iterated HODs has length at least η. We conclude by showing that the lengths of the sequences of inner mantles and of iterated HODs can be separated to be any two ordinals you please.
We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the grou... more We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the ground model to be the η-th inner mantle of the extension, in which the sequence of inner mantles has length at least η. This answers a conjecture of Fuchs, Hamkins, and Reitz [FHR15] in the positive. We also show that M(η) forces the ground model to be the η-th iterated HOD of the extension, where the sequence of iterated HODs has length at least η. We conclude by showing that the lengths of the sequences of inner mantles and of iterated HODs can be separated to be any two ordinals you please.
This paper explores how a pluralist view can arise in a natural way out of the day-today practice... more This paper explores how a pluralist view can arise in a natural way out of the day-today practice of modern set theory. By contrast, the widely accepted orthodox view is that there is an ultimate universe of sets V , and it is in this universe that mathematics takes place. From this view, the purpose of set theory is "learning the truth about V ." It has become apparent, however, that the phenomenon of independence-those questions left unresolved by the axioms-holds a central place in the investigation. This paper introduces the notion of independence, explores the primary tool ("soundness") for establishing independence results, and shows how a plurality of models arises through the investigation of this phenomenon. Building on a familiar example from Euclidean geometry, a template for independence proofs is established. Applying this template in the domain of set theory leads to a consideration of forcing, the tool par excellence for constructing universes of sets. Fifty years of forcing has resulted in a profusion of universes exhibiting a wide variety of characteristics-a multiverse of set theories. Direct study of this multiverse presents technical challenges due to its second-order nature. Nonetheless, there are certain nice "local neighborhoods" of the multiverse that are amenable to first-order analysis, and set-theoretic geology studies just such a neighborhood, the collection of grounds of a given universe V of set theory. I will explore some of the properties of this collection, touching on major concepts, open questions, and recent developments.
A pointwise definable model is one in which every object is definable without parameters. In a mo... more A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. 2000 Mathematics Subject Classification. 03E55. Key words and phrases. set theory, forcing. 1 See [Ani] for an instance of the argument at MathOverflow, which surely serves a brisk cup of math tea online. We leave aside the remark of Horatio, eight-year-old son of the first author, who announced, "Sure, papa, I can describe any number. Let me show you: tell me any number, and I'll tell you a description of it!"
In light of the celebrated theorem of Vopěnka [VH72], proving in ZFC that every set is generic ov... more In light of the celebrated theorem of Vopěnka [VH72], proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe V must be a class-forcing extension of HOD by some possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its HOD for any class forcing notion definable in HOD and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman [Fri12] showed, positively, that if one augments HOD with a certain ZFC-amenable class A, definable in V , then the set-theoretic universe V is a class-forcing extension of the expanded structure HOD, ∈, A. Our result shows that this augmentation process can be necessary. The same example shows that V is not necessarily a class-forcing extension of the mantle, and the method provides counterexamples to the intermediate model property, namely, a class-forcing extension V ⊆ W ⊆ V [G] with an intermediate transitive inner model W that is neither a class-forcing extension of V nor a ground model of V [G] by any definable class forcing notion with definable forcing relations.
Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a su... more Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a subset to κ by forcing: the Cohen poset Add(κ, 1), and the Cohen poset of the inner model Add(κ, 1) W. The forcing from W will be at least as strong as the forcing from V (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from V to fail to be as strong as that from W. The results are generalized to Add(κ, λ), and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.
Axiom is first-order expressible, and any model of zfc has a class-forcing extension which satisf... more Axiom is first-order expressible, and any model of zfc has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion v=hod that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing. First and foremost I would like to thank my advisor, Joel David Hamkins, an inspiration in mathematics and in life. He sets high standards for his students and conforms to even higher standards himself. An aspiring mathematician could not ask for a better mentor. Thank you also to my committee for their time, attention, and insightful comments. My close friends and respected colleagues Victoria Gitman and Thomas Johnstone, who have shared five years of intense and wonderful study with me, deserve much credit for enabling me to produce this work. Thank you for listening to my ideas and sharing your own. Learning has never been such an adventure and such a pleasure as it has been with you. Thank you to my family for instilling the crazy idea that I could accomplish whatever I wanted, and for teaching me about love. Finally, to my wife and best friend Gwen I owe the gratitude of uncomplaining support and unconditional love through far too many trials.
We introduce and consider the inner-model reflection principle, which asserts that whenever a sta... more We introduce and consider the inner-model reflection principle, which asserts that whenever a statement ϕ(a) in the first-order language of set theory is true in the set-theoretic universe V , then it is also true in a proper inner model W V. A stronger principle, the ground-model reflection principle, asserts that any such ϕ(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the innermodel hypothesis IMH. We also consider some questions concerning the expressibility of the principles.
We introduce and consider the inner-model reflection principle, which asserts that whenever a sta... more We introduce and consider the inner-model reflection principle, which asserts that whenever a statement ϕ(a) in the first-order language of set theory is true in the set-theoretic universe V , then it is also true in a proper inner model W V. A stronger principle, the ground-model reflection principle, asserts that any such ϕ(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π 2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.
A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC... more A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, wh...
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set fo... more A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent.
A pointwise definable model is one in which every object is definable without parameters. In a mo... more A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. 2000 Mathematics Subject Classification. 03E55. Key words and phrases. set theory, forcing. 1 See [Ani] for an instance of the argument at MathOverflow, which surely serves a brisk cup of math tea online. We leave aside the remark of Horatio, eight-year-old son of the first author, who announced, "Sure, papa, I can describe any number. Let me show you: tell me any number, and I'll tell you a description of it!"
A ground of the universe V is a transitive proper class W ⊆ V , such that W |= ZFC and V is obtai... more A ground of the universe V is a transitive proper class W ⊆ V , such that W |= ZFC and V is obtained by set forcing over W , so that V = W [G] for some W-generic filter G ⊆ P ∈ W. The model V satisfies the ground axiom GA if there are no such W properly contained in V. The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V. The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V. The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.
Abstract. The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of... more Abstract. The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, it is first-order expressible in set theory. The previously known models of the Ground Axiom all satisfy strong forms of V =HOD. In this article, we show that the Ground Axiom is relatively consistent with V ̸ = HOD. In fact, every model of ZFC has a class-forcing extension that is a model of ZFC + GA + V ̸ = HOD. The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a classforcing extension with ZFC + GA + V ̸ = HOD in which this supercompact cardinal is preserved. The Ground Axiom, introduced by Hamkins and Reitz [10, 9, 4], is the assertion that the universe of set theory is not a nontrivial set-forcing extension of any inner model. That is, the Ground Axiom asserts that if W is an inner model of the universe V and G is W-generic for nontrivial forcing, th...
In light of the celebrated theorem of Vop\v{e}nka (1972), proving in ZFC that every set is generi... more In light of the celebrated theorem of Vop\v{e}nka (1972), proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of HOD by some possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its HOD for any class forcing notion definable in HOD and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman (2012) showed, positively, that if one augments HOD with a certain ZFC-amenable class $A$, definable in $V$, then the set-theoretic universe $V$ is a class-forcing extension of the expanded structure $\langle\text{HOD},\in,A\rangle$. Our result shows that this augmentation process can be necessary. The same example shows that $V$ is not necessarily a class-forcing extension of the mantle, and the method provides counterexamples to the intermediate model property, n...
Axiom is first-order expressible, and any model of zfc has a class-forcing extension which satisf... more Axiom is first-order expressible, and any model of zfc has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion v=hod that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing. First and foremost I would like to thank my advisor, Joel David Hamkins, an inspiration in mathematics and in life. He sets high standards for his students and conforms to even higher standards himself. An aspiring mathematician could not ask for a better mentor. Thank you also to my committee for their time, attention, and insightful comments. My close friends and respected colleagues Victoria Gitman and Thomas Johnstone, who have shared five years of intense and wonderful study with me, deserve much credit for enabling me to produce this work. Thank you for listening to my ideas and sharing your own. Learning has never been such an adventure and such a pleasure as it has been with you. Thank you to my family for instilling the crazy idea that I could accomplish whatever I wanted, and for teaching me about love. Finally, to my wife and best friend Gwen I owe the gratitude of uncomplaining support and unconditional love through far too many trials.
Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a su... more Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a subset to κ by forcing: the Cohen poset Add(κ, 1), and the Cohen poset of the inner model Add(κ, 1) W . The forcing from W will be at least as strong as the forcing from V (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from V to fail to be as strong as that from W . The results are generalized to Add(κ, λ), and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.
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