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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Core models are inner
models.
The first core model, Dodd-Jensen core model ($K^{DJ}$), was introduced
in (Dodd & Jensen, 1981). The core model built
assuming
$¬ 0 ^{sword}$
is called *the core built using measures of order 0*
($K^{MOZ}$).(Sharpe & Welch, 2011) The core
model is often denoted $\mathbf{K}$.

(Further informations from (Dodd & Jensen, 1981))

Definition 6.3:

- $D = \{ \langle \xi, \kappa \rangle : \xi \in C_N, \text{$N$ is a mouse at $\kappa$}, |C_N| = \omega \}$
- $D_\alpha = \{ \langle \xi, \kappa \rangle : \xi \in C_N, \text{$N$ 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).

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.

- Dodd, A., & Jensen, R. (1981). The core model.
*Ann. Math. Logic*,*20*(1), 43–75. https://doi.org/10.1016/0003-4843(81)90011-5 - Sharpe, I., & Welch, P. (2011). Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties.
*Ann. Pure Appl. Logic*,*162*(11), 863–902. https://doi.org/10.1016/j.apal.2011.04.002