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
Quick navigation
The upper attic
The middle attic
The lower attic
The parlour
The playroom
The library
The cellar
Sources
Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
Latest working snapshot at the wayback machine
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:
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.