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 $\mathrm{PA}$
(<a href="index.php?title=Peano_arithmetic&action=edit&redlink=1" class="new" title="Peano arithmetic (page does not exist)">Peano arithmetic</a>)
(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
<a href="Transitive_ZFC_model" class="mw-redirect" title="Transitive ZFC model">Transitive ZFC model</a>.
## Class-sized transitive models
One can also talk about class-sized transitive models. Inner model is a
[transitive](Transitive.md "Transitive")
class (from other point of view, a class-sized transitive model (of ZFC
or a weaker theory)) containing all ordinals.
[Forcing](Forcing.md "Forcing")
creates outer models, but it can also be used in relation with inner
models.{% cite Fuchs2015 %}
Among them are *canonical inner models* like
- the [](Core%20models.md)
- the canonical model
[$L\[\mu\]$](Constructible%20universe.md "Constructible universe")
of one measurable cardinal
- [HOD](HOD.md "HOD") and
generic HOD (gHOD)
- mantle $\mathbb{M}$ (=generic mantle $g\mathbb{M}$)
- outer core
- the [](Constructible%20universe.md)
$L$
### 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. {% cite Fuchs2015 Usuba2017 %}
**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. {% cite Fuchs2015 Usuba2017 %}
**$α$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**.{% cite Fuchs2015 %}
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.).{% cite Fuchs2015 %}
For a cardinal $κ$, we call a ground $W$ of $V$ a $κ$-ground if there is
a poset $\mathbb{P} ∈ W$ of size
lt; κ$ and a $(W,
\mathbb{P})$-generic $G$ such that $V = W\[G\]$. The **$κ$-mantle** is
the intersection of all
$κ$-grounds.{% cite Usuba2019 %}
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.{% cite Usuba2019 %}
#### <span id="Mantle_and_large_cardinals" class="mw-headline">Mantle and large cardinals</span>
If $\kappa$ is
<a href="Hyperhuge" class="mw-redirect" title="Hyperhuge">hyperhuge</a>,
then $V$ has lt;\kappa$ many
<a href="Ground" class="mw-redirect" title="Ground">grounds</a>
(so the mantle is a ground
itself).{% cite Usuba2017 %}
If $κ$ is
[extendible](Extendible.md "Extendible")
then the $κ$-mantle of $V$ is its smallest ground (so of course the
mantle is a ground of
$V$).{% cite Usuba2019 %}
On the other hand, it s consistent that there is a
[supercompact](Supercompact.md "Supercompact")
cardinal and class many grounds of $V$ (because of the indestructibility
properties of
supercompactness).{% cite Usuba2017 %}
## $\kappa$-model
A **weak $κ$-model** is a
[transitive](Transitive.md "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 $M^{<\kappa} \subseteq
M$. {% cite Hamkins2014 Holy2018 %}