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
A model of a theory $T$ is a set $M$ together with relations (eg. two: $a$ and $b$) satisfying all axioms of the theory $T$. Symbolically $\langle M, a, b \rangle \models T$. According to the Gödel completeness theorem, in $\mathrm{PA}$ (Peano arithmetic) (so also in $\mathrm{ZFC}$) a theory has models iff it is consistent. According to Löwenheim–Skolem theorem, in $\mathrm{ZFC}$ if a countable first-order theory has an infinite model, it has infinite models of all cardinalities.
A model of a set theory (eg. $\mathrm{ZFC}$) is a set $M$ such that the structure $\langle M,\hat\in \rangle$ satisfies all axioms of the set theory. If $\hat \in$ is base theory’s $\in$, the model is called a transitive model. Gödel completeness theorem and Löwenheim–Skolem theorem do not apply to transitive models. (But Löwenheim–Skolem theorem together with Mostowski collapsing lemma show that if there is a transitive model of ZFC, then there is a countable such model.) See Transitive ZFC model.
One can also talk about class-sized transitive models. Inner model is a transitive class (from other point of view, a class-sized transitive model (of ZFC or a weaker theory)) containing all ordinals. Forcing creates outer models, but it can also be used in relation with inner models.(Fuchs et al., 2015)
Among them are canonical inner models like
The mantle $\mathbb{M}$ is the intersection of all grounds. Mantle is always a model of ZFC. Mantle is a ground (and is called a bedrock) iff $V$ has only set many grounds. (Fuchs et al., 2015; Usuba, 2017)
Generic mantle $g\mathbb{M}$ was defined as the intersection of all mantles of generic extensions, but then it turned out that it is identical to the mantle. (Fuchs et al., 2015; Usuba, 2017)
$α$th inner mantle $\mathbb{M}^α$ is defined by $\mathbb{M}^0=V$, $\mathbb{M}^{α+1} = \mathbb{M}^{\mathbb{M}^α}$ (mantle of the previous inner mantle) and $\mathbb{M}^α = \bigcap_{β<α} \mathbb{M}^β$ for limit $α$. If there is uniform presentation of $\mathbb{M}^α$ for all ordinals $α$ as a single class, one can talk about $\mathbb{M}^\mathrm{Ord}$, $\mathbb{M}^{\mathrm{Ord}+1}$ etc. If an inner mantle is a ground, it is called the outer core.(Fuchs et al., 2015)
It is conjenctured (unproved) that every model of ZFC is the $\mathbb{M}^α$ of another model of ZFC for any desired $α ≤ \mathrm{Ord}$, in which the sequence of inner mantles does not stabilise before $α$. It is probable that in the some time there are models of ZFC, for which inner mantle is undefined (Analogously, a 1974 result of Harrington appearing in (Zadrożny, 1983, section 7), with related work in (McAloon, 1974), shows that it is relatively consistent with Gödel-Bernays set theory that $\mathrm{HOD}^n$ exists for each $n < ω$ but the intersection $\mathrm{HOD}^ω = \bigcap_n \mathrm{HOD}^n$ is not a class.).(Fuchs et al., 2015)
For a cardinal $κ$, we call a ground $W$ of $V$ a $κ$-ground if there is a poset $\mathbb{P} ∈ W$ of size $< κ$ and a $(W, \mathbb{P})$-generic $G$ such that $V = W[G]$. The $κ$-mantle is the intersection of all $κ$-grounds.(Usuba, 2019)
The $κ$-mantle is a definable, transitive, and extensional class. It is consistent that the $κ$-mantle is a model of ZFC (e.g. when there are no grounds), and if $κ$ is a strong limit, then the $κ$-mantle must be a model of ZF. However it is not known whether the $κ$-mantle is always a model of ZFC.(Usuba, 2019)
If $\kappa$ is hyperhuge, then $V$ has $<\kappa$ many grounds (so the mantle is a ground itself).(Usuba, 2017)
If $κ$ is extendible then the $κ$-mantle of $V$ is its smallest ground (so of course the mantle is a ground of $V$).(Usuba, 2019)
On the other hand, it s consistent that there is a supercompact cardinal and class many grounds of $V$ (because of the indestructibility properties of supercompactness).(Usuba, 2017)
A weak $κ$-model is a transitive set $M$ of size $\kappa$ with $\kappa \in M$ and satisfying the theory $\mathrm{ZFC}^-$ ($\mathrm{ZFC}$ without the axiom of power set, with collection, not replacement). It is a $κ$-model if additionaly $M^{<\kappa} \subseteq M$. (Hamkins & Johnstone, 2014; Holy & Schlicht, 2018)