Return to Question

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\phi_i\}_{i<\alpha}$ to $ZFC$ and produce a new theory $T=ZFC+\{\phi_i\}_{i<n}$ with $\mathcal{N}$ a model of $T$ such that $\mathcal{M}$ is a transitive model of $ZFC$ in $\mathcal{N}$ not satisfying $\{\phi_i\}_{i<n}$, we might* obtain new cardinals and ordinals in $\mathcal{N}$ not in $\mathcal{M}$ which in turn yield new surreal numbers in $N_0^\mathcal{N}$.

Has this been studied at all/is it obvious to someone here how to deal with this when trying to study the surreals? In particular I wonder about new algebraic/topological/categorical/(someday) analytical properties of $N_0^\mathcal{N}$ in comparison with $N_0^\mathcal{M}$.

For example, if $T=ZFC+\text{there exists a weakly inaccessible cardinal}$ then $N_0^\mathcal{N}$ has a natural value class "larger" than $N_0^\mathcal{M}$ since each new inaccessible introduced yields a new possible value. As mentioned below the model theory of $N_0$ when only paying attention to its arithmetic operations is characterized entirely by it being real-closed, but a value class is additional data and Andreas Blass brings up the excellent question of what additional data on the surreals would allow them to interpret the whole set theoretical universe.

I'm currently worrying about much smaller cardinals (equivalent to Grothendieck universes) for the purpose of letting the Surreals be an object in the category ${\bf CRing}$ -- note here that not only is the class of objects in ${\bf CRing}$ a proper class, some of the objects themselves are also proper classes in an 'uninitiated' universe.

It seems that we're 'changing' the surreals when we introduce the standard additional large cardinal axioms used in category theory for dealing with proper classes; is this true, and if so how can we understand the changes in a precise manner?

My intuition says that there is an answer along the lines of 'restrict to a certain size surreals and use those', but ideally I would like to work with a version of the surreals containing all known large cardinals to see what they look like, so a more general methodology is desirable.

*as was pointed out in the comments by Andrés E. Caicedo and Noah Schweber, it is possible for $\mathcal{N}$ to retain all ordinals and lose cardinals in this situation if the submodel $\mathcal{M}$ has fewer surjections between small ordinals and larger ones, but I'm not sure the surreals 'care' whether an ordinal is a cardinal or not.

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\phi_i\}_{i<\alpha}$ to $ZFC$ and produce a new theory $T=ZFC+\{\phi_i\}_{i<n}$ with $\mathcal{N}$ a model of $T$ such that $\mathcal{M}$ is a transitive model of $ZFC$ in $\mathcal{N}$ not satisfying $\{\phi_i\}_{i<n}$, we might* obtain new cardinals and ordinals in $\mathcal{N}$ not in $\mathcal{M}$ which in turn yield new surreal numbers in $N_0^\mathcal{N}$.

Has this been studied at all/is it obvious to someone here how to deal with this when trying to study the surreals? In particular I wonder about new algebraic/topological/categorical/(someday) analytical properties of $N_0^\mathcal{N}$ in comparison with $N_0^\mathcal{M}$.

For example, if $T=ZFC+\text{there exists a weakly inaccessible cardinal}$ then $N_0^\mathcal{N}$ has a natural value class "larger" than $N_0^\mathcal{M}$ since each new inaccessible introduced yields a new possible value. As mentioned below the model theory of $N_0$ when only paying attention to its arithmetic operations is characterized entirely by it being real-closed, but a value class is additional data and Andreas Blass brings up the excellent question of what additional data on the surreals would allow them to interpret the whole set theoretical universe.

I'm currently worrying about much smaller cardinals (equivalent to Grothendieck universes) for the purpose of letting the Surreals be an object in the category ${\bf CRing}$ -- note here that not only is the class of objects in ${\bf CRing}$ a proper class, some of the objects themselves are also proper classes in an 'uninitiated' universe.

It seems that we're 'changing' the surreals when we introduce the standard additional large cardinal axioms used in category theory for dealing with proper classes; is this true, and if so how can we understand the changes in a precise manner?

My intuition says that there is an answer along the lines of 'restrict to a certain size surreals and use those', but ideally I would like to work with a version of the surreals containing all known large cardinals to see what they look like, so a more general methodology is desirable.

*as was pointed out in the comments by Andrés E. Caicedo and Noah Schweber, it is possible for $\mathcal{N}$ to retain all ordinals and lose cardinals in this situation if the submodel $\mathcal{M}$ is an inner model and has fewer surjections between small ordinals and larger ones, but I'm not sure the surreals 'care' whether an ordinal is a cardinal or not.

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\phi_i\}_{i<\alpha}$ to $ZFC$ and produce a new theory $T=ZFC+\{\phi_i\}_{i<n}$ with $\mathcal{N}$ a model of $T$ such that $\mathcal{M}$ is a transitive model of $ZFC$ in $\mathcal{N}$ not satisfying $\{\phi_i\}_{i<n}$, we might* obtain new cardinals and ordinals in $\mathcal{N}$ not in $\mathcal{M}$ which in turn yield new surreal numbers in $N_0^\mathcal{N}$.

*see Andrés E. Caicedo's comment below, it is possible retain all ordinals and lose cardinals in this situation

Has this been studied at all/is it obvious to someone here how to deal with this when trying to study the surreals? In particular I wonder about new algebraic/topological/categorical/(someday) analytical properties of $N_0^\mathcal{N}$ in comparison with $N_0^\mathcal{M}$.

For example, I would imagine that if $T=ZFC+\text{there exists a supercompact cardinal}$ then $N_0^\mathcal{N}$ has a natural value class "larger" than $N_0^\mathcal{M}$ since each new inaccessible introduced yields a new possible value. As mentioned below the model theory of $N_0$ when only paying attention to its arithmetic operations is characterized entirely by it being real-closed, but a value class is additional data and Andreas Blass brings up the excellent question of what additional data on the surreals would allow them to interpret the whole set theoretical universe.

I'm currently worrying about much smaller cardinals (equivalent to Grothendieck universes) for the purpose of letting the Surreals be an object in the category ${\bf CRing}$ -- note here that not only is the class of objects in ${\bf CRing}$ a proper class, some of the objects themselves are also proper classes in an 'uninitiated' universe.

It seems that we're 'changing' the surreals when we introduce the standard additional large cardinal axioms used in category theory for dealing with proper classes; is this true, and if so how can we understand the changes in a precise manner?

My intuition says that there is an answer along the lines of 'restrict to a certain size surreals and use those', but ideally I would like to work with a version of the surreals containing all known large cardinals to see what they look like, so a more general methodology is desirable.

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\phi_i\}_{i<\alpha}$ to $ZFC$ and produce a new theory $T=ZFC+\{\phi_i\}_{i<n}$ with $\mathcal{N}$ a model of $T$ such that $\mathcal{M}$ is a transitive model of $ZFC$ in $\mathcal{N}$ not satisfying $\{\phi_i\}_{i<n}$, we might* obtain new cardinals and ordinals in $\mathcal{N}$ not in $\mathcal{M}$ which in turn yield new surreal numbers in $N_0^\mathcal{N}$.

Has this been studied at all/is it obvious to someone here how to deal with this when trying to study the surreals? In particular I wonder about new algebraic/topological/categorical/(someday) analytical properties of $N_0^\mathcal{N}$ in comparison with $N_0^\mathcal{M}$.

For example, if $T=ZFC+\text{there exists a weakly inaccessible cardinal}$ then $N_0^\mathcal{N}$ has a natural value class "larger" than $N_0^\mathcal{M}$ since each new inaccessible introduced yields a new possible value. As mentioned below the model theory of $N_0$ when only paying attention to its arithmetic operations is characterized entirely by it being real-closed, but a value class is additional data and Andreas Blass brings up the excellent question of what additional data on the surreals would allow them to interpret the whole set theoretical universe.

I'm currently worrying about much smaller cardinals (equivalent to Grothendieck universes) for the purpose of letting the Surreals be an object in the category ${\bf CRing}$ -- note here that not only is the class of objects in ${\bf CRing}$ a proper class, some of the objects themselves are also proper classes in an 'uninitiated' universe.

It seems that we're 'changing' the surreals when we introduce the standard additional large cardinal axioms used in category theory for dealing with proper classes; is this true, and if so how can we understand the changes in a precise manner?

My intuition says that there is an answer along the lines of 'restrict to a certain size surreals and use those', but ideally I would like to work with a version of the surreals containing all known large cardinals to see what they look like, so a more general methodology is desirable.

*as was pointed out in the comments by Andrés E. Caicedo and Noah Schweber, it is possible for $\mathcal{N}$ to retain all ordinals and lose cardinals in this situation if the submodel $\mathcal{M}$ has fewer surjections between small ordinals and larger ones, but I'm not sure the surreals 'care' whether an ordinal is a cardinal or not.

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\phi_i\}_{i<\alpha}$ to $ZFC$ and produce a new theory $T=ZFC+\{\phi_i\}_{i<n}$ with $\mathcal{N}$ a model of $T$ such that $\mathcal{M}$ is a transitive model of $ZFC$ in $\mathcal{N}$ not satisfying $\{\phi_i\}_{i<n}$, we might* obtain new cardinals and ordinals in $\mathcal{N}$ not in $\mathcal{M}$ which in turn yield new surreal numbers in $N_0^\mathcal{N}$.

*see Andrés E. Caicedo's comment below, apparently we may retail all ordinals and lose cardinals

Has this been studied at all/is it obvious to someone here how to deal with this when trying to study the surreals? In particular I wonder about new algebraic/topological/categorical/(someday) analytical properties of $N_0^\mathcal{N}$ in comparison with $N_0^\mathcal{M}$.

For example, I would imagine that if $T=ZFC+\text{there exists a supercompact cardinal}$ then $N_0^\mathcal{N}$ has a natural value class "larger" than $N_0^\mathcal{M}$ since each new inaccessible introduced yields a new possible value. As mentioned below the model theory of $N_0$ when only paying attention to its arithmetic operations is characterized entirely by it being real-closed, but a value class is additional data and Andreas Blass brings up the excellent question of what additional data on the surreals would allow them to interpret the whole set theoretical universe.

I'm currently worrying about much smaller cardinals (equivalent to Grothendieck universes) for the purpose of letting the Surreals be an object in the category ${\bf CRing}$ -- note here that not only is the class of objects in ${\bf CRing}$ a proper class, some of the objects themselves are also proper classes in an 'uninitiated' universe.

It seems that we're 'changing' the surreals when we introduce the standard additional large cardinal axioms used in category theory for dealing with proper classes; is this true, and if so how can we understand the changes in a precise manner?

My intuition says that there is an answer along the lines of 'restrict to a certain size surreals and use those', but ideally I would like to work with a version of the surreals containing all known large cardinals to see what they look like, so a more general methodology is desirable.

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\phi_i\}_{i<\alpha}$ to $ZFC$ and produce a new theory $T=ZFC+\{\phi_i\}_{i<n}$ with $\mathcal{N}$ a model of $T$ such that $\mathcal{M}$ is a transitive model of $ZFC$ in $\mathcal{N}$ not satisfying $\{\phi_i\}_{i<n}$, we might* obtain new cardinals and ordinals in $\mathcal{N}$ not in $\mathcal{M}$ which in turn yield new surreal numbers in $N_0^\mathcal{N}$.

*see Andrés E. Caicedo's comment below, it is possible retain all ordinals and lose cardinals in this situation

Has this been studied at all/is it obvious to someone here how to deal with this when trying to study the surreals? In particular I wonder about new algebraic/topological/categorical/(someday) analytical properties of $N_0^\mathcal{N}$ in comparison with $N_0^\mathcal{M}$.

For example, I would imagine that if $T=ZFC+\text{there exists a supercompact cardinal}$ then $N_0^\mathcal{N}$ has a natural value class "larger" than $N_0^\mathcal{M}$ since each new inaccessible introduced yields a new possible value. As mentioned below the model theory of $N_0$ when only paying attention to its arithmetic operations is characterized entirely by it being real-closed, but a value class is additional data and Andreas Blass brings up the excellent question of what additional data on the surreals would allow them to interpret the whole set theoretical universe.

I'm currently worrying about much smaller cardinals (equivalent to Grothendieck universes) for the purpose of letting the Surreals be an object in the category ${\bf CRing}$ -- note here that not only is the class of objects in ${\bf CRing}$ a proper class, some of the objects themselves are also proper classes in an 'uninitiated' universe.

It seems that we're 'changing' the surreals when we introduce the standard additional large cardinal axioms used in category theory for dealing with proper classes; is this true, and if so how can we understand the changes in a precise manner?

My intuition says that there is an answer along the lines of 'restrict to a certain size surreals and use those', but ideally I would like to work with a version of the surreals containing all known large cardinals to see what they look like, so a more general methodology is desirable.