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For about two years I have been in contact with a splinter group of amateur enthusiasts who have been striving towards this endeavor. One of their considered objects is "an infinity so large, that it cannot be realized as a von-Neumann-style collection of smaller objects of a single sort". I will attempt to justify that the concept of set is flexible enough that (1) not only is multi-sortedness not necessary for unlocking any novel structure, but also that, post-adjoinment, such objects cannot even be considered "too large to be sets" (2).

The main motive behind the endeavor of defining infinities past $\mathsf{Ord}$ seems to often be the idea that set-hood is a restrictive property, since realizability as a collection of objects of a single sort is something that has to be shed when considering objects built from proper classes like $\textrm{Ord}$. However, behavior of two-sorted models can be emulated by a one-sorted model via a simple translation based on, e.g. $(V,W,\in)\mapsto (\{0\}\times V\cup\{1\}\times W,\in^*)$, where $a\in^*b$ iff $\exists x,y(a=(0,x)\land x\in y)$. Since behavior of proper classes can be emulated via sets, we shouldn't expect novel behavior to appear when considering objects larger than $\textrm{V}$, e.g. the behavior of proper classes can be couched in terms of a "restructuring" of $(V,\in)$, and no special two-sortedness is needed.

So there is no novel behavior whose existence necessitates abandoning one-sortedness of the universe, for example if we were to adjoin a larger-than-Ord object $\Omega$ to the universe, the apparent behavior of the new $\mathsf{Ord}$ can be couched in terms of a "reordering" of the original $\mathsf{Ord}$. But can we even call the adjoined objects "large"? For example, if we have a model $(M,\in)$ of ZFC, and we adjoin an ordinal $\Omega\notin M$ to $M$, post-adjoinment can we even say that $\Omega$ is "larger than $\mathsf{Ord}$"? $\Omega$ was indeed not a member of Ord^M, the collection $\{\alpha\in M\mid(M,\in)\vDash``\alpha\textrm{ is an ordinal}"\}$. However, after adjoining $\Omega$, even in a structure as small as $(M\cup\{\Omega\},\in)$ we will have $(M\cup\{\Omega\},\in)\vDash``\Omega\textrm{ is an ordinal}"$, therefore $\Omega$ is a member of $\mathsf{Ord}^{M\cup\{\Omega\}}$ and in this extension $\Omega$ is not larger than $\mathsf{Ord}$. The same argument holds for adjoinment of non-ordinal $\Omega$, if all preceding instances of "$\mathsf{Ord}$" are replaced with "$V$".

For about two years I have been in contact with a splinter group of amateur enthusiasts who have been striving towards this endeavor. One of their considered objects is "an infinity so large, that it cannot be realized as a von-Neumann-style collection of smaller objects of a single sort". I will attempt to justify that the concept of set is flexible enough that (1) not only is multi-sortedness not necessary for unlocking any novel structure, but also that, post-adjoinment, such objects cannot even be considered "too large to be sets" (2).

Edit: This section was edited in June 2023 following a comment by user Holo.

The main motive behind the endeavor of defining infinities past $\mathsf{Ord}$ seems to often be the idea that set-hood is a restrictive property, since realizability as a collection of objects of a single sort is something that has to be shed when considering objects built from proper classes like $\textrm{Ord}$. However, behavior of two-sorted models can be emulated by a one-sorted model via a simple translation based on, e.g. $(V,W,\in)\mapsto (\{0\}\times V\cup\{1\}\times W,\in^*)$, where $a\in^*b$ iff $\exists x,y(a=(0,x)\land x\in y)$. Since behavior of proper classes can be emulated via sets, we shouldn't expect novel behavior to appear when considering objects larger than $\textrm{V}$, e.g. the behavior of proper classes can be couched in terms of a "restructuring" of $(V,\in)$, and no special two-sortedness is needed.

So there is no novel behavior whose existence necessitates abandoning one-sortedness of the universe, for example if we were to adjoin a larger-than-Ord object $\Omega$ to the universe, the apparent behavior of the new $\mathsf{Ord}$ can be couched in terms of a "reordering" of the original $\mathsf{Ord}$. But can we even call the adjoined objects "large"? For example, if we have a model $(M,\in)$ of ZFC, and we adjoin an ordinal $\Omega\notin M$ to $M$, post-adjoinment can we even say that $\Omega$ is "larger than $\mathsf{Ord}$"? $\Omega$ was indeed not a member of Ord^M, the collection $\{\alpha\in M\mid(M,\in)\vDash``\alpha\textrm{ is an ordinal}"\}$. However, after adjoining $\Omega$, even in a structure as small as $(M\cup\{\Omega\},\in)$ we will have $(M\cup\{\Omega\},\in)\vDash``\Omega\textrm{ is an ordinal}"$, therefore $\Omega$ is a member of $\mathsf{Ord}^{M\cup\{\Omega\}}$ and in this extension $\Omega$ is not larger than $\mathsf{Ord}$. The same argument holds for adjoinment of non-ordinal $\Omega$, if all preceding instances of "$\mathsf{Ord}$" are replaced with "$V$".

For about two years I have been in contact with a splinter group of amateur enthusiasts who have been striving towards this endeavor. One of their considered objects is "an infinity so large, that it cannot be realized as a von-Neumann-style collection of smaller objects of a single sort". I will attempt to justify that the concept of set is flexible enough that (1) not only does considering adjoinment of "objects larger than $V$" not unlock any novel structure, but also that, post-adjoinment, such objects cannot even be considered "too large to be sets" (2).

The main motive behind the endeavor of defining infinities past $\mathsf{Ord}$ seems to often be the idea that set-hood is a restrictive property, like finiteness or countablility. But in view of the maximal iterative conception of set, being a set is not a restriction on the size of an object, it only requires that the object is an extensional well-founded collection with a rank.

Here are two examples that should show that restrictive properties like finiteness have more baggage attached than set-hood. Suppose we have the set $H_{\aleph_0}$ of hereditarily finite sets, and we are working over ZFC-Powerset in the model $(H_{\aleph_0},\in)$. If we were to take some infinite object $\Omega\notin M$, then adjoin $\Omega$ to $M$ and form some new model of ZFC-Powerset $(M',\in)$ with $M'\supseteq M\cup\{\Omega\}$, we have that $(M',\in)\vDash``\Omega\textrm{ is infinite}"$. From the perspective of the original model $(M,\in)$, this behavior called "infiniteness" exemplified by this new element $\Omega$ is novel. Because finiteness is a restrictive assumption, something novel is gained by "going past it". The same argument applies for some other restrictive properties, like countability, equinumerosity with $\mathbb R$, etc.

However, if we have a model $(M,\in)$ of ZFC and an $\Omega\notin M$, then in a model extending $M\cup\{\Omega\}$, $\Omega$ is still a set. This is in contrast to the much more concrete properties of finiteness or countability, which have a "reason" to fail for particular objects $\Omega$ upon their adjoinment to a model (lack of a certain bijection). In particular there is no choice of $M,\Omega$ such that $\Omega$ is not a set in the extension, so there is no such "novel behavior beyond set-hood" obtained by adding a large object to a model.

So "post-set" behavior is not introduced by adjoining "larger than $V$" objects to the universe. But can we even call the adjoined objects "large"? For example, if we have a model $(M,\in)$ of ZFC, and we adjoin an ordinal $\Omega\notin M$ to $M$, post-adjoinment can we even say that $\Omega$ is "larger than $\mathsf{Ord}$"? $\Omega$ was indeed not a member of Ord^M, the collection $\{\alpha\in M\mid(M,\in)\vDash``\alpha\textrm{ is an ordinal}"\}$. However, after adjoining $\Omega$, even in a structure as small as $(M\cup\{\Omega\},\in)$ we will have $(M\cup\{\Omega\},\in)\vDash``\Omega\textrm{ is an ordinal}"$, therefore $\Omega$ is a member of $\mathsf{Ord}^{M\cup\{\Omega\}}$ and in this extension $\Omega$ is not larger than $\mathsf{Ord}$. The same argument holds for adjoinment of non-ordinal $\Omega$, if all preceding instances of "$\mathsf{Ord}$" are replaced with "$V$".

For about two years I have been in contact with a splinter group of amateur enthusiasts who have been striving towards this endeavor. One of their considered objects is "an infinity so large, that it cannot be realized as a von-Neumann-style collection of smaller objects of a single sort". I will attempt to justify that the concept of set is flexible enough that (1) not only is multi-sortedness not necessary for unlocking any novel structure, but also that, post-adjoinment, such objects cannot even be considered "too large to be sets" (2).

The main motive behind the endeavor of defining infinities past $\mathsf{Ord}$ seems to often be the idea that set-hood is a restrictive property, since realizability as a collection of objects of a single sort is something that has to be shed when considering objects built from proper classes like $\textrm{Ord}$. However, behavior of two-sorted models can be emulated by a one-sorted model via a simple translation based on, e.g. $(V,W,\in)\mapsto (\{0\}\times V\cup\{1\}\times W,\in^*)$, where $a\in^*b$ iff $\exists x,y(a=(0,x)\land x\in y)$. Since behavior of proper classes can be emulated via sets, we shouldn't expect novel behavior to appear when considering objects larger than $\textrm{V}$, e.g. the behavior of proper classes can be couched in terms of a "restructuring" of $(V,\in)$, and no special two-sortedness is needed.

So there is no novel behavior whose existence necessitates abandoning one-sortedness of the universe, for example if we were to adjoin a larger-than-Ord object $\Omega$ to the universe, the apparent behavior of the new $\mathsf{Ord}$ can be couched in terms of a "reordering" of the original $\mathsf{Ord}$. But can we even call the adjoined objects "large"? For example, if we have a model $(M,\in)$ of ZFC, and we adjoin an ordinal $\Omega\notin M$ to $M$, post-adjoinment can we even say that $\Omega$ is "larger than $\mathsf{Ord}$"? $\Omega$ was indeed not a member of Ord^M, the collection $\{\alpha\in M\mid(M,\in)\vDash``\alpha\textrm{ is an ordinal}"\}$. However, after adjoining $\Omega$, even in a structure as small as $(M\cup\{\Omega\},\in)$ we will have $(M\cup\{\Omega\},\in)\vDash``\Omega\textrm{ is an ordinal}"$, therefore $\Omega$ is a member of $\mathsf{Ord}^{M\cup\{\Omega\}}$ and in this extension $\Omega$ is not larger than $\mathsf{Ord}$. The same argument holds for adjoinment of non-ordinal $\Omega$, if all preceding instances of "$\mathsf{Ord}$" are replaced with "$V$".

For about two years I have been in contact with a splinter group of amateur enthusiasts who have been striving towards this endeavor. One of their considered objects is "an infinity so large, that it cannot be realized as a von-Neumann-style collection of smaller objects of a single sort". I will attempt to justify that (1) not only does considering adjoinment of "objects larger than $V$" not unlock any novel structure, but also that, post-adjoinment, such objects cannot even be considered "too large to be sets" (2).

The main motive behind the endeavor of defining infinities past $\mathsf{Ord}$ seems to often be the idea that set-hood is a restrictive property, like finiteness or countablility. But in view of the maximal iterative conception of set, being a set is not a restriction on the size of an object, it only requires that the object is an extensional well-founded collection with a rank.

Here are two examples that should show that restrictive properties like finiteness have more baggage attached than set-hood. Suppose we have the set $H_{\aleph_0}$ of hereditarily finite sets, and we are working over ZFC-Powerset in the model $(H_{\aleph_0},\in)$. If we were to take some infinite object $\Omega\notin M$, then adjoin $\Omega$ to $M$ and form some new model of ZFC-Powerset $(M',\in)$ with $M'\supseteq M\cup\{\Omega\}$, we have that $(M',\in)\vDash``\Omega\textrm{ is infinite}"$. From the perspective of the original model $(M,\in)$, this behavior called "infiniteness" exemplified by this new element $\Omega$ is novel. Because finiteness is a restrictive assumption, something novel is gained by "going past it". The same argument applies for some other restrictive properties, like countability, equinumerosity with $\mathbb R$, etc.

However, if we have a model $(M,\in)$ of ZFC and an $\Omega\notin M$, then in a model extending $M\cup\{\Omega\}$, $\Omega$ is still a set. This is in contrast to the much more concrete properties of finiteness or countability, which have a "reason" to fail for particular objects $\Omega$ upon their adjoinment to a model (lack of a certain bijection). In particular there is no choice of $M,\Omega$ such that $\Omega$ is not a set in the extension, so there is no such "novel behavior beyond set-hood" obtained by adding a large object to a model.

So "post-set" behavior is not introduced by adjoining "larger than $V$" objects to the universe. But can we even call the adjoined objects "large"? For example, if we have a model $(M,\in)$ of ZFC, and we adjoin an ordinal $\Omega\notin M$ to $M$, post-adjoinment can we even say that $\Omega$ is "larger than $\mathsf{Ord}$"? $\Omega$ was indeed not a member of Ord^M, the collection $\{\alpha\in M\mid(M,\in)\vDash``\alpha\textrm{ is an ordinal}"\}$. However, after adjoining $\Omega$, even in a structure as small as $(M\cup\{\Omega\},\in)$ we will have $(M\cup\{\Omega\},\in)\vDash``\Omega\textrm{ is an ordinal}"$, therefore $\Omega$ is a member of $\mathsf{Ord}^{M\cup\{\Omega\}}$ and in this extension $\Omega$ is not larger than $\mathsf{Ord}$. The same argument holds for adjoinment of non-ordinal $\Omega$, if all preceding instances of "$\mathsf{Ord}$" are replaced with "$V$".

For about two years I have been in contact with a splinter group of amateur enthusiasts who have been striving towards this endeavor. One of their considered objects is "an infinity so large, that it cannot be realized as a von-Neumann-style collection of smaller objects of a single sort". I will attempt to justify that the concept of set is flexible enough that (1) not only does considering adjoinment of "objects larger than $V$" not unlock any novel structure, but also that, post-adjoinment, such objects cannot even be considered "too large to be sets" (2).

The main motive behind the endeavor of defining infinities past $\mathsf{Ord}$ seems to often be the idea that set-hood is a restrictive property, like finiteness or countablility. But in view of the maximal iterative conception of set, being a set is not a restriction on the size of an object, it only requires that the object is an extensional well-founded collection with a rank.

Here are two examples that should show that restrictive properties like finiteness have more baggage attached than set-hood. Suppose we have the set $H_{\aleph_0}$ of hereditarily finite sets, and we are working over ZFC-Powerset in the model $(H_{\aleph_0},\in)$. If we were to take some infinite object $\Omega\notin M$, then adjoin $\Omega$ to $M$ and form some new model of ZFC-Powerset $(M',\in)$ with $M'\supseteq M\cup\{\Omega\}$, we have that $(M',\in)\vDash``\Omega\textrm{ is infinite}"$. From the perspective of the original model $(M,\in)$, this behavior called "infiniteness" exemplified by this new element $\Omega$ is novel. Because finiteness is a restrictive assumption, something novel is gained by "going past it". The same argument applies for some other restrictive properties, like countability, equinumerosity with $\mathbb R$, etc.

However, if we have a model $(M,\in)$ of ZFC and an $\Omega\notin M$, then in a model extending $M\cup\{\Omega\}$, $\Omega$ is still a set. This is in contrast to the much more concrete properties of finiteness or countability, which have a "reason" to fail for particular objects $\Omega$ upon their adjoinment to a model (lack of a certain bijection). In particular there is no choice of $M,\Omega$ such that $\Omega$ is not a set in the extension, so there is no such "novel behavior beyond set-hood" obtained by adding a large object to a model.

So "post-set" behavior is not introduced by adjoining "larger than $V$" objects to the universe. But can we even call the adjoined objects "large"? For example, if we have a model $(M,\in)$ of ZFC, and we adjoin an ordinal $\Omega\notin M$ to $M$, post-adjoinment can we even say that $\Omega$ is "larger than $\mathsf{Ord}$"? $\Omega$ was indeed not a member of Ord^M, the collection $\{\alpha\in M\mid(M,\in)\vDash``\alpha\textrm{ is an ordinal}"\}$. However, after adjoining $\Omega$, even in a structure as small as $(M\cup\{\Omega\},\in)$ we will have $(M\cup\{\Omega\},\in)\vDash``\Omega\textrm{ is an ordinal}"$, therefore $\Omega$ is a member of $\mathsf{Ord}^{M\cup\{\Omega\}}$ and in this extension $\Omega$ is not larger than $\mathsf{Ord}$. The same argument holds for adjoinment of non-ordinal $\Omega$, if all preceding instances of "$\mathsf{Ord}$" are replaced with "$V$".