# Core models
Core models are inner [models](Model.md "Model").
The first core model, [[Dodd-Jensen core model]] ($K^{DJ}$), was introduced in {% cite Dodd1981 %}.
The core model built assuming $¬ 0 ^{sword}$ (see [[Zero sword]]) is called *the [[Core model for measures of order 0]]* ($K^{MOZ}$).{% cite Sharpe2011 %}
The core model is often denoted $\mathbf{K}$.
(Further informations from {% cite Dodd1981 %})
## From the definition
Definition 6.3:
- $D = \{ \langle \xi, \kappa \rangle \mid \xi \in C_N, N\text{ is a mouse at }$\kappa$, |C_N| = \omega \}$
- $D_\alpha = \{ \langle \xi, \kappa \rangle \mid \xi \in C_N, N \text{ is a mouse at }\kappa, \|C_N\| = \omega, \mathrm{Ord} \cap N < \omega_\alpha \}$
- $K = L[D]$ — **the core model**
- $K_\alpha = |J_\alpha^D|$
Definition 5.4: $N$ is a *mouse* iff $N$ is a critical premouse, $N'$ is
iterable and for each $i \in \mathrm{Ord}$ there is $N_i$, a critical
premouse, such that $(N_i)' = N_i'$ where $\langle N_i',
\pi_{ij}', \kappa_i \rangle$ is the iteration of $N'$, and $n(N_i)= n(N)$.
Definition 5.1: Premouse $N = J_\alpha^U$ is *critical* iff
$\mathcal{P}(\kappa) \cap \Sigma_\omega(N) \not\subseteq N$ and
$N$ is acceptable.
Definition 3.1: For $\kappa < \alpha$, $N = J_\alpha^U$ is a
*premouse* at $\kappa$ iff $N \models
quot;$U$ is a normal measure"
on $\kappa$”}$.
$J_\alpha^A$ is defined using functions rudimentary in $A$
(definitions 1.1, 1.2).
## Properties
The core model $K$ is not absolute, for example: if $0^\sharp$ ([[Zero sharp]]) does not exist, then $K = L$; if $0^\sharp$ exists, but $0^{\sharp\sharp}$
does not, then $K = L[0^\sharp]$.
However, $K^M = M \cap K$ for any inner model $M$.
$K$ will contain “all the sharps” in the universe, but may in general be larger than the model obtained by iterating the $\sharp$ operation
through the ordinals.