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

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The upper attic
The middle attic
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Sources
Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
Latest working snapshot at the wayback machine

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:

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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