The Wholeness Axioms
The wholeness axioms, proposed by Paul Corazza [1, 2], occupy a high place in the upper stratosphere of the large cardinal hierarchy, intended as slight weakenings of the Kunen inconsistency, but similar in spirit.
The wholeness axioms are formalized in the language $\{\in,j\}$, augmenting the usual language of set theory $\{\in\}$ with an additional unary function symbol $j$ to represent an elementary embedding from $V$ into itself, with higher and higher degrees of seperation in $\{\in,j\}$.
Contents
Wholeness axioms
The base theory ZFC is expressed only in the smaller language $\{\in\}$. Corazza's original proposal, which we denote by $\textrm{WA}_0$, asserts that $j$ is a nontrivial amenable elementary embedding from the universe to itself, without adding formulas containing $j$ to the separation and replacement axioms. Elementarity is expressed by the scheme $\varphi(x)\iff\varphi(j(x))$, where $\varphi$ runs through the formulas of the usual language of set theory; nontriviality is expressed by the sentence $\exists x j(x)\not=x$; and amenability is simply the assertion that $j\upharpoonright A$ is a set for every set $A$. Amenability in this case is equivalent to the assertion that the separation axiom holds for $\Delta_0$ formulae in the language $\{\in,j\}$. The wholeness axiom $\textrm{WA}$, also denoted $\textrm{WA}_\infty$, asserts in addition that the full separation axiom holds in the language $\{\in,j\}$.
Those two axioms are the endpoints of the hierarchy of axioms $\textrm{WA}_n$, asserting increasing amounts of the separation axiom. Specifically, the wholeness axiom $\textrm{WA}_n$, where $n$ is amongst $0,1,\ldots,\infty$, consists of the following:
- (Elementarity) All instances of $\varphi(x)\iff\varphi(j(x))$ for $\varphi$ in the language $\{\in,j\}$.
- (Separation) All instances of the Separation Axiom for $\Sigma_n$ formulae in the full language $\{\in,j\}$.
- (Nontriviality) The axiom $\exists x\,j(x)\not=x$.
Clearly, this resembles the Kunen inconsistency. What is missing from the wholeness axiom schemes, and what figures prominantly in Kunen's proof, are the instances of the replacement axiom in the full language with $j$. In particular, it is the replacement axiom in the language with $j$ that allows one to define the critical sequence $\langle \kappa_n\mid n\lt\omega\rangle$, where $\kappa_{n+1}=j(\kappa_n)$, which figures in all the proofs of the Kunen inconsistency. Thus, none of the proofs of the Kunen inconsistency can be carried out with $\textrm{WA}$, and indeed, in every model of $\textrm{WA}$ the critical sequence is unbounded in the ordinals.
Axioms $\mathrm{I}_4^n$
(Section from [2])
Axioms $\mathrm{I}_4^n$ for natural numbers $n$ (Starting from $0$) are an attempt to measure the gap between $\mathrm{I}_3$ and $\mathrm{WA}$. Each of these axioms asserts the existence of a transitive model of $\mathrm{ZFC} + \mathrm{WA}$ with additional, stronger and stronger properties. Namely, $\mathrm{I}_4^n(\kappa)$ holds if and only if there is a transitive model $(M,\in,j)$ of $\mathrm ZFC+WA$ with $V_{j^n(\kappa)+1}\subseteq M$ and $\kappa$ the critical point of $j$.
It is not known whether the $\mathrm{I}_4^n$ axioms really increase in consistency strength and whether it is possible in $\mathrm{ZFC}$ that $\forall _{n\in\omega} \mathrm{I}_4^n(\kappa) \land\neg\mathrm{I}_3(\kappa)$.
A sequence of elementary embeddings $\langle i_n: M_n\prec M_n|n\lt\omega\rangle$ is called $\mathrm{I}_4(\kappa)$-coherent if and only if for each $n$:
- $i_n$ has critical point $\kappa$,
- $\langle M_n,\in,i_n\rangle$ witnesses $\mathrm{I}_4^n(\kappa)$,
- and for each $m\lt n$ $i_m\restriction V_{\kappa_{(m)}}=i_n\restriction V_{\kappa_{(m)}}$, where $\kappa_{(0)}=\kappa$ and $\kappa_{(m)}=i_m^m(\kappa)$.
$\mathrm{BTEE}$ etc.
(Section from [3] unless noted otherwise)
The Basic Theory of Elementary Embeddings $\mathrm{BTEE}$ states Elementarity (formulated above) and Critical Point (“There is a least ordinal moved by $j$”).
Cofinal Axiom states that $∀_α ∃_{n ∈ ω} (j^n(\kappa)$ exists and $α ≤ j^n(\kappa))$ (The critical sequence is cofinal in the ordinals).[2]
Similar theories can have very different consistency strengths.
Properties
- If the wholeness axiom is consistent with $\textrm{ZFC}$, then it is consistent with $\textrm{ZFC+V=HOD}$.[4]
- The hierarchy of wholeness axioms is strictly increasing, if consistent.[4]
- If $\mathrm{I}_4^0(\kappa)$, then $\kappa$ is measurable and $\{\alpha\lt\kappa|\alpha\textrm{ is measurable}\}$ has measure 1.[2]
- If $\mathrm{I}_4^1(\kappa)$, then $\kappa$ is superstrong and $\{\alpha\lt\kappa|\alpha\textrm{ is superstrong}\}$ has measure 1.[2]
- If $n\ge 1$ and $\mathrm{I}_4^{n+1}(\kappa)$, then $\kappa$ is $n$-huge and $\{\alpha\lt\kappa|\alpha\textrm{ is }n\textrm{-huge}\}$ has measure 1.[2]
- If $i$ witnesses $\mathrm{I}_4^n(\kappa)$, then $i^n(\kappa)$ is measurable.[2]
- $\mathrm{I}_3(\kappa)$ is equivalent to the existence of an $\mathrm{I}_4(\kappa)$-coherent set of embeddings.[2]
- If there is some $j: V_\lambda\prec V_\lambda$, then $\langle V_\lambda,\in,j \rangle\vDash WA$.
- $\mathrm{ZFC} + \mathrm{WA}_0 + \Sigma_0\textrm{-Collection}_\mathbf{j}$ is inconsistent.[3]
- $\mathrm{ZFC} + \mathrm{WA}_0$ proves the Cofinal Axiom.[2]
- If there is an $\omega$-Erdős cardinal, than there is a transitive set model of $\mathrm{ZFC} + \mathrm{BTEE}$.[3]
- Consistency of a completely ineffable cardinal implies consistency of the theory $\mathrm{BTEE}$.[1]
- For each particular natural number in the metatheory $n ≥ 1$, $\mathrm{ZFC} + \mathrm{BTEE}$ proves that the critical point of $j$ is $n$-ineffable.[3]
- $\mathrm{ZFC} + \mathrm{BTEE}$ proves that the critical point of $j$ is totally indescribable.[3]
- If $\mathrm{ZFC}$ is consistent, then there is a model $\langle M, E, j \rangle$ of $\mathrm{ZFC} + \text{Elementarity} + \text{Nontriviality}$.[3]
- However, a well founded model of this theory must also satisfy Critical Point and hence $\mathrm{BTEE}$.
References
- Corazza, Paul. The Wholeness Axiom and Laver sequences. Annals of Pure and Applied Logic pp. 157--260, October, 2000. bibtex
- Corazza, Paul. The gap between $\mathrm{I}_3$ and the wholeness axiom. Fund Math 179(1):43--60, 2003. www DOI MR bibtex
- Corazza, Paul. The spectrum of elementary embeddings $j : V \to V$. Annals of Pure and Applied Logic 139(1--3):327-399, May, 2006. DOI bibtex
- Hamkins, Joel David. The wholeness axioms and V=HOD. Arch Math Logic 40(1):1--8, 2001. www arχiv DOI MR bibtex