Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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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
$\langle M, a, b \rangle \models T$. 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 $\langle M,\hat\in \rangle$ 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.

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 $L$

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)

**$α$th inner mantle** $\mathbb{M}^α$ is defined by $\mathbb{M}^0=V$,
$\mathbb{M}^{α+1} = \mathbb{M}^{\mathbb{M}^α}$ (mantle of the
previous inner mantle) and $\mathbb{M}^α = \bigcap_{β<α}
\mathbb{M}^β$ for limit $α$. If there is uniform presentation of
$\mathbb{M}^α$ for all ordinals $α$ 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 $\mathbb{M}^α$ of another model of ZFC for any desired $α ≤ \mathrm{Ord}$, in which the sequence of inner mantles does not stabilise before $α$. 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 $n < ω$ but the intersection $\mathrm{HOD}^ω = \bigcap_n \mathrm{HOD}^n$ is not a class.).(Fuchs et al., 2015)

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

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

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

If $κ$ is extendible then the $κ$-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)

A **weak $κ$-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 **$κ$-model** if
additionaly $M^{<\kappa} \subseteq
M$. (Hamkins & Johnstone, 2014; Holy & Schlicht, 2018)

- 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 - 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 - 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 - Hamkins, J. D., & Johnstone, T. A. (2014).
*Resurrection axioms and uplifting cardinals*. http://jdh.hamkins.org/resurrection-axioms-and-uplifting-cardinals/ - Holy, P., & Schlicht, P. (2018). A hierarchy of Ramsey-like cardinals.
*Fundamenta Mathematicae*,*242*, 49–74. https://doi.org/10.4064/fm396-9-2017