Model

• Class-sized transitive models
  • Mantle
    • Mantle and large cardinals
• $\kappa$-model

A model of a theory $T$ is a set $M$ together with relations (eg. two: $a$ and $b$) satisfying all axioms of the theory $T$. Symbolically $⟨M,a,b⟩\models T$. According to the Gödel completeness theorem, in $\mathrm{PA}$ (Peano arithmetic) (so also in $\mathrm{ZFC}$) a theory has models iff it is consistent. According to Löwenheim–Skolem theorem, in $\mathrm{ZFC}$ if a countable first-order theory has an infinite model, it has infinite models of all cardinalities.

A model of a set theory (eg. $\mathrm{ZFC}$) is a set $M$ such that the structure $⟨M,\hat{\in }⟩$ satisfies all axioms of the set theory. If $\hat{\in }$ is base theory’s $\in$, the model is called a transitive model. Gödel completeness theorem and Löwenheim–Skolem theorem do not apply to transitive models. (But Löwenheim–Skolem theorem together with Mostowski collapsing lemma show that if there is a transitive model of ZFC, then there is a countable such model.) See Transitive ZFC model.

Class-sized transitive models

One can also talk about class-sized transitive models. Inner model is a transitive class (from other point of view, a class-sized transitive model (of ZFC or a weaker theory)) containing all ordinals. Forcing creates outer models, but it can also be used in relation with inner models.(Fuchs et al., 2015)

Among them are canonical inner models like

• the core model
• the canonical model $L[\mu ]$ of one measurable cardinal
• HOD and generic HOD (gHOD)
• mantle $\mathbb{M}$ (=generic mantle $g\mathbb{M}$)
• outer core
• the constructible universe $L$

Mantle

The mantle $\mathbb{M}$ is the intersection of all grounds. Mantle is always a model of ZFC. Mantle is a ground (and is called a bedrock) iff $V$ has only set many grounds. (Fuchs et al., 2015; Usuba, 2017)

Generic mantle $g\mathbb{M}$ was defined as the intersection of all mantles of generic extensions, but then it turned out that it is identical to the mantle. (Fuchs et al., 2015; Usuba, 2017)

$\alpha$th inner mantle ${\mathbb{M}}^{\alpha }$ is defined by ${\mathbb{M}}^0=V$, ${\mathbb{M}}^{\alpha +1}={\mathbb{M}}^{{\mathbb{M}}^{\alpha }}$ (mantle of the previous inner mantle) and ${\mathbb{M}}^{\alpha }=\bigcap _{\beta <\alpha }{\mathbb{M}}^{\beta }$ for limit $\alpha$. If there is uniform presentation of ${\mathbb{M}}^{\alpha }$ for all ordinals $\alpha$ as a single class, one can talk about ${\mathbb{M}}^{\mathrm{Ord}}$, ${\mathbb{M}}^{\mathrm{Ord}+1}$ etc. If an inner mantle is a ground, it is called the outer core.(Fuchs et al., 2015)

It is conjenctured (unproved) that every model of ZFC is the ${\mathbb{M}}^{\alpha }$ of another model of ZFC for any desired $\alpha \le \mathrm{Ord}$, in which the sequence of inner mantles does not stabilise before $\alpha$. It is probable that in the some time there are models of ZFC, for which inner mantle is undefined (Analogously, a 1974 result of Harrington appearing in (Zadrożny, 1983, section 7), with related work in (McAloon, 1974), shows that it is relatively consistent with Gödel-Bernays set theory that ${\mathrm{HOD}}^n$ exists for each $n<\omega$ but the intersection ${\mathrm{HOD}}^{\omega }=\bigcap _n{\mathrm{HOD}}^n$ is not a class.).(Fuchs et al., 2015)

For a cardinal $\kappa$, we call a ground $W$ of $V$ a $\kappa$-ground if there is a poset $\mathbb{P}\in W$ of size $<\kappa$ and a $(W,\mathbb{P})$-generic $G$ such that $V=W[G]$. The $\kappa$-mantle is the intersection of all $\kappa$-grounds.(Usuba, 2019)

The $\kappa$-mantle is a definable, transitive, and extensional class. It is consistent that the $\kappa$-mantle is a model of ZFC (e.g. when there are no grounds), and if $\kappa$ is a strong limit, then the $\kappa$-mantle must be a model of ZF. However it is not known whether the $\kappa$-mantle is always a model of ZFC.(Usuba, 2019)

Mantle and large cardinals

If $\kappa$ is hyperhuge, then $V$ has $<\kappa$ many grounds (so the mantle is a ground itself).(Usuba, 2017)

If $\kappa$ is extendible then the $\kappa$-mantle of $V$ is its smallest ground (so of course the mantle is a ground of $V$).(Usuba, 2019)

On the other hand, it s consistent that there is a supercompact cardinal and class many grounds of $V$ (because of the indestructibility properties of supercompactness).(Usuba, 2017)

$\kappa$-model

A weak $\kappa$-model is a transitive set $M$ of size $\kappa$ with $\kappa \in M$ and satisfying the theory ${\mathrm{ZFC}}^{-}$ ($\mathrm{ZFC}$ without the axiom of power set, with collection, not replacement). It is a $\kappa$-model if additionaly $M^{<\kappa }\subseteq M$. (Hamkins & Johnstone, 2014; Holy & Schlicht, 2018)

References

1. Fuchs, G., Hamkins, J. D., & Reitz, J. (2015). Set-theoretic geology. Annals of Pure and Applied Logic, 166(4), 464–501. https://doi.org/http://web.archive.org/web/20191116153209/https://doi.org/10.1016/j.apal.2014.11.004
2. Usuba, T. (2017). The downward directed grounds hypothesis and very large cardinals. Journal of Mathematical Logic, 17(02), 1750009. https://doi.org/10.1142/S021906131750009X
3. Usuba, T. (2019). Extendible cardinals and the mantle. Archive for Mathematical Logic, 58(1-2), 71–75. https://doi.org/10.1007/s00153-018-0625-4
4. Hamkins, J. D., & Johnstone, T. A. (2014). Resurrection axioms and uplifting cardinals. http://jdh.hamkins.org/resurrection-axioms-and-uplifting-cardinals/
5. Holy, P., & Schlicht, P. (2018). A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae, 242, 49–74. https://doi.org/10.4064/fm396-9-2017

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