SLIDE 1
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
A naturalistic justification of the Generic Introduction Multiverse - - PowerPoint PPT Presentation
A naturalistic justification of the Generic Introduction Multiverse - - PowerPoint PPT Presentation
A justification of the GM H M. de Ceglie A naturalistic justification of the Generic Introduction Multiverse with a core Background The main argument Conclusions Matteo de Ceglie decegliematteo@gmail.com Paris Lodron Universitt Salzburg
SLIDE 2
SLIDE 3
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Structure of the Presentation
1 Introduction 2 Background 3 The main argument 4 Conclusions
SLIDE 4
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Structure of the Presentation
1 Introduction 2 Background 3 The main argument 4 Conclusions
SLIDE 5
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Structure of the Presentation
1 Introduction 2 Background 3 The main argument 4 Conclusions
SLIDE 6
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Structure of the Presentation
1 Introduction 2 Background 3 The main argument 4 Conclusions
SLIDE 7
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Introduction outline
Pluralism, Anti-Pluralism and Naturalism The reasons of the emergence of the multiverse:
Independent propositions; Alternative set theories.
A brief sketch of the main argument
SLIDE 8
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Introduction outline
Pluralism, Anti-Pluralism and Naturalism The reasons of the emergence of the multiverse:
Independent propositions; Alternative set theories.
A brief sketch of the main argument
SLIDE 9
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Introduction outline
Pluralism, Anti-Pluralism and Naturalism The reasons of the emergence of the multiverse:
Independent propositions; Alternative set theories.
A brief sketch of the main argument
SLIDE 10
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Introduction outline
Pluralism, Anti-Pluralism and Naturalism The reasons of the emergence of the multiverse:
Independent propositions; Alternative set theories.
A brief sketch of the main argument
SLIDE 11
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Introduction outline
Pluralism, Anti-Pluralism and Naturalism The reasons of the emergence of the multiverse:
Independent propositions; Alternative set theories.
A brief sketch of the main argument
SLIDE 12
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Pluralism, Anti-Pluralism and Naturalism
Naturalism Mathematical practice should be considered the final judge for questions in philosophy of mathematics. Anti-Pluralism There is only one set theoretic universe. Pluralism There are various set theoretic universes.
SLIDE 13
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Pluralism, Anti-Pluralism and Naturalism
Naturalism Mathematical practice should be considered the final judge for questions in philosophy of mathematics. Anti-Pluralism There is only one set theoretic universe. Pluralism There are various set theoretic universes.
SLIDE 14
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Pluralism, Anti-Pluralism and Naturalism
Naturalism Mathematical practice should be considered the final judge for questions in philosophy of mathematics. Anti-Pluralism There is only one set theoretic universe. Pluralism There are various set theoretic universes.
SLIDE 15
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
A brief Sketch of the argument
The multiverse is just as good, when dealing with actual mathematical practice, as the single universe; Moreover, in the multiverse is possible to prove more things than in the single universe; Thus, from a naturalistic point of view, the multiverse should be preferred over the single universe.
SLIDE 16
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
A brief Sketch of the argument
The multiverse is just as good, when dealing with actual mathematical practice, as the single universe; Moreover, in the multiverse is possible to prove more things than in the single universe; Thus, from a naturalistic point of view, the multiverse should be preferred over the single universe.
SLIDE 17
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
A brief Sketch of the argument
The multiverse is just as good, when dealing with actual mathematical practice, as the single universe; Moreover, in the multiverse is possible to prove more things than in the single universe; Thus, from a naturalistic point of view, the multiverse should be preferred over the single universe.
SLIDE 18
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Structure of the Presentation
1 Introduction 2 Background 3 The main argument 4 Conclusions
SLIDE 19
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Background outline
The Multiverse conceptions in set theory
The broad multiverse; The Generic Multiverse with a core (GMH)
The naturalistic approach
SLIDE 20
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Background outline
The Multiverse conceptions in set theory
The broad multiverse; The Generic Multiverse with a core (GMH)
The naturalistic approach
SLIDE 21
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Background outline
The Multiverse conceptions in set theory
The broad multiverse; The Generic Multiverse with a core (GMH)
The naturalistic approach
SLIDE 22
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Background outline
The Multiverse conceptions in set theory
The broad multiverse; The Generic Multiverse with a core (GMH)
The naturalistic approach
SLIDE 23
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Multiverse conceptions in set theory
The broad multiverse All the possible universes are part of the multiverse, with no hierarchy nor criterion to sort them. The generic multiverses In this kind of multiverses we differentiate between universes using a strong logic (an idea owed to Woodin, from now on GMΩ) or supposing the existence of a core (an idea owed to Steel, that is the GMH). The hyperverse A multiverse of multiverses.
SLIDE 24
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Multiverse conceptions in set theory
The broad multiverse All the possible universes are part of the multiverse, with no hierarchy nor criterion to sort them. The generic multiverses In this kind of multiverses we differentiate between universes using a strong logic (an idea owed to Woodin, from now on GMΩ) or supposing the existence of a core (an idea owed to Steel, that is the GMH). The hyperverse A multiverse of multiverses.
SLIDE 25
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Multiverse conceptions in set theory
The broad multiverse All the possible universes are part of the multiverse, with no hierarchy nor criterion to sort them. The generic multiverses In this kind of multiverses we differentiate between universes using a strong logic (an idea owed to Woodin, from now on GMΩ) or supposing the existence of a core (an idea owed to Steel, that is the GMH). The hyperverse A multiverse of multiverses.
SLIDE 26
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The Generic Multiverse with a core (GMH)
Definition of the core The core of the multiverse is the collection of all the statements that are true in every universe of the
- multiverse. We can then consider
every universe of the generic multiverse and extension of its core.
SLIDE 27
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The naturalistic approach
The problem Which is best for mathematical practice? The Single Universe framework or the Multiverse?. UNIFY Our framework should be foundational. MAXIMIZE The framework theory should be as powerful as possible, not restricting in any way the development of the foundations of mathematics (the framework theory should be the most Generous Arena for mathematics).
SLIDE 28
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The naturalistic approach
The problem Which is best for mathematical practice? The Single Universe framework or the Multiverse?. UNIFY Our framework should be foundational. MAXIMIZE The framework theory should be as powerful as possible, not restricting in any way the development of the foundations of mathematics (the framework theory should be the most Generous Arena for mathematics).
SLIDE 29
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The naturalistic approach
The problem Which is best for mathematical practice? The Single Universe framework or the Multiverse?. UNIFY Our framework should be foundational. MAXIMIZE The framework theory should be as powerful as possible, not restricting in any way the development of the foundations of mathematics (the framework theory should be the most Generous Arena for mathematics).
SLIDE 30
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Some more about the naturalistic approach
We could further refine UNIFY defining the following foundationality feature:
Meta-mathematical Corral; Elucidation; Shared Standard; Risk Assessment.
A candidate framework for mathematical practice should at least provide all these features.
SLIDE 31
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Some more about the naturalistic approach
We could further refine UNIFY defining the following foundationality feature:
Meta-mathematical Corral; Elucidation; Shared Standard; Risk Assessment.
A candidate framework for mathematical practice should at least provide all these features.
SLIDE 32
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Some more about the naturalistic approach
We could further refine UNIFY defining the following foundationality feature:
Meta-mathematical Corral; Elucidation; Shared Standard; Risk Assessment.
A candidate framework for mathematical practice should at least provide all these features.
SLIDE 33
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Some more about the naturalistic approach
We could further refine UNIFY defining the following foundationality feature:
Meta-mathematical Corral; Elucidation; Shared Standard; Risk Assessment.
A candidate framework for mathematical practice should at least provide all these features.
SLIDE 34
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Some more about the naturalistic approach
We could further refine UNIFY defining the following foundationality feature:
Meta-mathematical Corral; Elucidation; Shared Standard; Risk Assessment.
A candidate framework for mathematical practice should at least provide all these features.
SLIDE 35
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Some more about the naturalistic approach
We could further refine UNIFY defining the following foundationality feature:
Meta-mathematical Corral; Elucidation; Shared Standard; Risk Assessment.
A candidate framework for mathematical practice should at least provide all these features.
SLIDE 36
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Structure of the Presentation
1 Introduction 2 Background 3 The main argument 4 Conclusions
SLIDE 37
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Main argument outline
The foundationality of the GMH Maximizing the descriptive power of the GMH The non-foundationality of the other multiverse conceptions
SLIDE 38
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Main argument outline
The foundationality of the GMH Maximizing the descriptive power of the GMH The non-foundationality of the other multiverse conceptions
SLIDE 39
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Main argument outline
The foundationality of the GMH Maximizing the descriptive power of the GMH The non-foundationality of the other multiverse conceptions
SLIDE 40
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The foundationality of the GMH
The multiverse core provides us all the foundationality feature needed to satisfy UNIFY:
Meta-mathematical Corrall; Elucidation; Shared Standard; Risk Assessment.
Thus, we can say that the GMH and the Single Universe are just as good. Given this, there would be no reason to switch from the Single Universe to the GMH.
SLIDE 41
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The foundationality of the GMH
The multiverse core provides us all the foundationality feature needed to satisfy UNIFY:
Meta-mathematical Corrall; Elucidation; Shared Standard; Risk Assessment.
Thus, we can say that the GMH and the Single Universe are just as good. Given this, there would be no reason to switch from the Single Universe to the GMH.
SLIDE 42
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The foundationality of the GMH
The multiverse core provides us all the foundationality feature needed to satisfy UNIFY:
Meta-mathematical Corrall; Elucidation; Shared Standard; Risk Assessment.
Thus, we can say that the GMH and the Single Universe are just as good. Given this, there would be no reason to switch from the Single Universe to the GMH.
SLIDE 43
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The foundationality of the GMH
The multiverse core provides us all the foundationality feature needed to satisfy UNIFY:
Meta-mathematical Corrall; Elucidation; Shared Standard; Risk Assessment.
Thus, we can say that the GMH and the Single Universe are just as good. Given this, there would be no reason to switch from the Single Universe to the GMH.
SLIDE 44
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The foundationality of the GMH
The multiverse core provides us all the foundationality feature needed to satisfy UNIFY:
Meta-mathematical Corrall; Elucidation; Shared Standard; Risk Assessment.
Thus, we can say that the GMH and the Single Universe are just as good. Given this, there would be no reason to switch from the Single Universe to the GMH.
SLIDE 45
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The foundationality of the GMH
The multiverse core provides us all the foundationality feature needed to satisfy UNIFY:
Meta-mathematical Corrall; Elucidation; Shared Standard; Risk Assessment.
Thus, we can say that the GMH and the Single Universe are just as good. Given this, there would be no reason to switch from the Single Universe to the GMH.
SLIDE 46
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
The foundationality of the GMH
The multiverse core provides us all the foundationality feature needed to satisfy UNIFY:
Meta-mathematical Corrall; Elucidation; Shared Standard; Risk Assessment.
Thus, we can say that the GMH and the Single Universe are just as good. Given this, there would be no reason to switch from the Single Universe to the GMH.
SLIDE 47
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Maximizing the power of the GMH
Lets suppose that our multiverse is composed by only two universes: one is a model of ZFC and the other a model of ZF + AD; In the multiverse, we retain all the results and true statements of ZFC and all the results of ZF + AD; Moreover, we can also prove several more interesting isomorphisms in this simplified multiverse; On the other hand, in the Single Universe, we limit
- urselves to only a subset of all the results we can prove in
the multiverse; Thus, considering MAXIMIZE, the multiverse is actually better than the Single Universe V .
SLIDE 48
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Maximizing the power of the GMH
Lets suppose that our multiverse is composed by only two universes: one is a model of ZFC and the other a model of ZF + AD; In the multiverse, we retain all the results and true statements of ZFC and all the results of ZF + AD; Moreover, we can also prove several more interesting isomorphisms in this simplified multiverse; On the other hand, in the Single Universe, we limit
- urselves to only a subset of all the results we can prove in
the multiverse; Thus, considering MAXIMIZE, the multiverse is actually better than the Single Universe V .
SLIDE 49
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Maximizing the power of the GMH
Lets suppose that our multiverse is composed by only two universes: one is a model of ZFC and the other a model of ZF + AD; In the multiverse, we retain all the results and true statements of ZFC and all the results of ZF + AD; Moreover, we can also prove several more interesting isomorphisms in this simplified multiverse; On the other hand, in the Single Universe, we limit
- urselves to only a subset of all the results we can prove in
the multiverse; Thus, considering MAXIMIZE, the multiverse is actually better than the Single Universe V .
SLIDE 50
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Maximizing the power of the GMH
Lets suppose that our multiverse is composed by only two universes: one is a model of ZFC and the other a model of ZF + AD; In the multiverse, we retain all the results and true statements of ZFC and all the results of ZF + AD; Moreover, we can also prove several more interesting isomorphisms in this simplified multiverse; On the other hand, in the Single Universe, we limit
- urselves to only a subset of all the results we can prove in
the multiverse; Thus, considering MAXIMIZE, the multiverse is actually better than the Single Universe V .
SLIDE 51
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Maximizing the power of the GMH
Lets suppose that our multiverse is composed by only two universes: one is a model of ZFC and the other a model of ZF + AD; In the multiverse, we retain all the results and true statements of ZFC and all the results of ZF + AD; Moreover, we can also prove several more interesting isomorphisms in this simplified multiverse; On the other hand, in the Single Universe, we limit
- urselves to only a subset of all the results we can prove in
the multiverse; Thus, considering MAXIMIZE, the multiverse is actually better than the Single Universe V .
SLIDE 52
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Against the other multiverse conceptions
All the other multiverse conceptions are equally powerful from the MAXIMIZE point of view; Although, they all fail the foundationality test:
The broad multiverse fails to provide Shared Standard and Risk Assessment; Woodin’s GMΩ fails to provide us Meta-mathematical Corrall; The hyperverse has the same problems of the broad multiverse, and moreover, cannot provide us with Elucidation.
SLIDE 53
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Against the other multiverse conceptions
All the other multiverse conceptions are equally powerful from the MAXIMIZE point of view; Although, they all fail the foundationality test:
The broad multiverse fails to provide Shared Standard and Risk Assessment; Woodin’s GMΩ fails to provide us Meta-mathematical Corrall; The hyperverse has the same problems of the broad multiverse, and moreover, cannot provide us with Elucidation.
SLIDE 54
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Against the other multiverse conceptions
All the other multiverse conceptions are equally powerful from the MAXIMIZE point of view; Although, they all fail the foundationality test:
The broad multiverse fails to provide Shared Standard and Risk Assessment; Woodin’s GMΩ fails to provide us Meta-mathematical Corrall; The hyperverse has the same problems of the broad multiverse, and moreover, cannot provide us with Elucidation.
SLIDE 55
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Against the other multiverse conceptions
All the other multiverse conceptions are equally powerful from the MAXIMIZE point of view; Although, they all fail the foundationality test:
The broad multiverse fails to provide Shared Standard and Risk Assessment; Woodin’s GMΩ fails to provide us Meta-mathematical Corrall; The hyperverse has the same problems of the broad multiverse, and moreover, cannot provide us with Elucidation.
SLIDE 56
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Against the other multiverse conceptions
All the other multiverse conceptions are equally powerful from the MAXIMIZE point of view; Although, they all fail the foundationality test:
The broad multiverse fails to provide Shared Standard and Risk Assessment; Woodin’s GMΩ fails to provide us Meta-mathematical Corrall; The hyperverse has the same problems of the broad multiverse, and moreover, cannot provide us with Elucidation.
SLIDE 57
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Structure of the Presentation
1 Introduction 2 Background 3 The main argument 4 Conclusions
SLIDE 58
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Conclusions
To conclude, we can say that the GMH is our best candidate to be the framework for mathematical practice:
It is as foundational as the classic set theoretic framework; Moreover, is the only multiverse conception that can claim to be foundational; It proves more isomorphisms than the classical set theoretic framework; Thus, from a naturalistic point of view, our only option is to accept that the GMH is better for mathematical practice than the classical set theoretic framework.
SLIDE 59
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Conclusions
To conclude, we can say that the GMH is our best candidate to be the framework for mathematical practice:
It is as foundational as the classic set theoretic framework; Moreover, is the only multiverse conception that can claim to be foundational; It proves more isomorphisms than the classical set theoretic framework; Thus, from a naturalistic point of view, our only option is to accept that the GMH is better for mathematical practice than the classical set theoretic framework.
SLIDE 60
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Conclusions
To conclude, we can say that the GMH is our best candidate to be the framework for mathematical practice:
It is as foundational as the classic set theoretic framework; Moreover, is the only multiverse conception that can claim to be foundational; It proves more isomorphisms than the classical set theoretic framework; Thus, from a naturalistic point of view, our only option is to accept that the GMH is better for mathematical practice than the classical set theoretic framework.
SLIDE 61
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions
Conclusions
To conclude, we can say that the GMH is our best candidate to be the framework for mathematical practice:
It is as foundational as the classic set theoretic framework; Moreover, is the only multiverse conception that can claim to be foundational; It proves more isomorphisms than the classical set theoretic framework; Thus, from a naturalistic point of view, our only option is to accept that the GMH is better for mathematical practice than the classical set theoretic framework.
SLIDE 62
A justification
- f the GMH
- M. de Ceglie
Introduction Background The main argument Conclusions