Model

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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; also in theories containing $\mathrm{PA}$, like $\mathrm{ZFC}$) 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 and Heights of models.

Class-sized transitive models

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.[1]

Among them are canonical inner models like

Some properties usually obtained by forcing are possible in inner models, for example[2]:

  • (theorem 14) If there is a supercompact cardinal, then there are inner models with an indestructible supercompact cardinal $κ$ such that
    • $2^κ = κ^+$
    • $2^κ = κ^{++}$
    • Moreover, for every cardinal $θ$, such inner models $W$ can be found for which also $W^θ ⊆ W$.

Mantle

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.[1, 3]

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.[1, 3]

$α$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.[1]

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.).[1]

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.[4]

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.[4]

Mantle and large cardinals

If $\kappa$ is hyperhuge, then $V$ has $<\kappa$ many grounds (so the mantle is a ground itself).[3]

If $κ$ is extendible then the $κ$-mantle of $V$ is its smallest ground (so of course the mantle is a ground of $V$).[4]

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).[3]

$\kappa$-model

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 $M^{<\kappa} \subseteq M$.[5, 6]

Prime models and minimal models

(from [7], p. 23-24 unless noted otherwise)

A minimal model is one without proper elementary submodels.

A prime model is one that embeds elementarily into every model of its theory. (compare [8], p. 4)

In general:

  • First order theories need not have either prime or minimal models.
  • Prime models need not be minimal, and minimal models need not be prime.

However, for a model $\mathfrak{M} \models \text{ZF}$, $\mathfrak{M}$ is a prime model $\implies$ $\mathfrak{M}$ is a Paris model and satisfies AC $\implies$ $\mathfrak{M}$ is a minimal model.

  • Neither implication reverses in general, but both do if $\mathfrak{M} \models V=HOD$.

The minimal transitive model of ZFC is an important model.

Solovay model

(from [9])

......

Definition: $L(\mathbb{R})^M$ is a Solovay model over $V$ for $V⊆M$ and $M$ satisfying: $\forall_{x∈\mathbb{R}}$ $ω_1$ is an inaccessible cardinal in $V[x]$ and $x$ is small-generic over $V$ (there is a forcing notion $\mathbb{P}$ in $V$ countable in $M$ and there is, in $M$, a $\mathbb{P}$-generic filter $g$ over $V$ such that $x∈V[g]$).

......

References

  1. Fuchs, Gunter and Hamkins, Joel David and Reitz, Jonas. Set-theoretic geology. Annals of Pure and Applied Logic 166(4):464 - 501, 2015. www   arχiv   DOI   bibtex
  2. Apter, Arthur W and Gitman, Victoria and Hamkins, Joel David. Inner models with large cardinal features usually obtained by forcing. Arch Math Logic 51(3-4):257--283, may, 2012. arχiv   bibtex
  3. Usuba, Toshimichi. The downward directed grounds hypothesis and very large cardinals. Journal of Mathematical Logic 17(02):1750009, 2017. arχiv   DOI   bibtex
  4. Usuba, Toshimichi. Extendible cardinals and the mantle. Archive for Mathematical Logic 58(1-2):71-75, 2019. arχiv   DOI   bibtex
  5. Hamkins, Joel David and Johnstone, Thomas A. Strongly uplifting cardinals and the boldface resurrection axioms. , 2014. arχiv   bibtex
  6. Holy, Peter and Schlicht, Philipp. A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae 242:49-74, 2018. www   arχiv   DOI   bibtex
  7. Enayat, Ali. Models of set theory with definable ordinals. Archive for Mathematical Logic 44:363–385, April, 2005. www   DOI   bibtex
  8. Hamkins, Joel David; Linetsky, David; Reitz, Jonas. Pointwise Definable Models of Set Theory. , 2012. arχiv   bibtex
  9. Bagaria, Joan and Bosch, Roger. Proper forcing extensions and Solovay models. Archive for Mathematical Logic , 2004. www   DOI   bibtex
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