cantors-attic

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.

View the Project on GitHub neugierde/cantors-attic

Core model

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))

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, A., & Jensen, R. (1981). The core model. Ann. Math. Logic, 20(1), 43–75. https://doi.org/10.1016/0003-4843(81)90011-5
2. 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
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