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 <a href="Zero_sword" class="mw-redirect" title="Zero sword">$¬ 0 ^{sword}lt;/a> is called *the core built using measures of order 0* ($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 $N \models \text{“$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 $0^\sharp$ 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.