Core model

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Core models are inner models. The first core model, Dodd-Jensen core model ($K^{DJ}$), was introduced in [1]. The core model built assuming $¬ 0 ^{sword}$ is called the core built using measures of order 0 ($K^{MOZ}$).[2] The core model is often denoted $\mathbf{K}$.

(Further informations from [1])

From the definition

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).

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.

References

  1. Dodd, Anthony and Jensen, Ronald. The core model. Ann Math Logic 20(1):43--75, 1981. www   DOI   MR   bibtex
  2. Sharpe, Ian and Welch, Philip. Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann Pure Appl Logic 162(11):863--902, 2011. www   DOI   MR   bibtex
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