Difference between revisions of "Reinhardt"

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The existence of '''Reinhardt cardinals''' has been refuted in $\text{ZFC}_2$ and $\text{GBC}$ by Kunen ([[Kunen inconsistency]]), the term is used in the $\text{ZF}_2$ context, although some mathematicians suspect that they are inconsistent even there.
 
The existence of '''Reinhardt cardinals''' has been refuted in $\text{ZFC}_2$ and $\text{GBC}$ by Kunen ([[Kunen inconsistency]]), the term is used in the $\text{ZF}_2$ context, although some mathematicians suspect that they are inconsistent even there.
  
== Definitions ==
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==Definitions==
  
A '''weakly Reinhardt cardinal'''(1) is the critical point $\kappa$ of a nontrivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$ such that $V_\kappa V$ ($\mathrm{WR}(\kappa)$. Existence of $\kappa$ is Weak Reinhardt Axiom ($\mathrm{WRA}$) by Woodin).<cite>Corazza2010:TheAxiomOfInfinityAndJVV</cite>:p.58
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A '''weakly Reinhardt cardinal'''(1) is the critical point $\kappa$ of a nontrivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$ such that $V_\kappa\prec V$ ($\mathrm{WR}(\kappa)$. Existence of $\kappa$ is Weak Reinhardt Axiom ($\mathrm{WRA}$) by Woodin).<cite>Corazza2010:TheAxiomOfInfinityAndJVV</cite>:p.58
  
A '''weakly Reinhardt cardinal'''(2) is the critical point $\kappa$ of a nontrivial elementary embedding $j:V_{\lambda+2}\to V_{\lambda+2}$ such that $V_\kappa V_\lambda V_\gamma$ (for some $\gamma > \lambda > \kappa$).<cite>Baaz2011:Kurt</cite>:(definition 20.6, p. 455)
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A '''weakly Reinhardt cardinal'''(2) is the critical point $\kappa$ of a nontrivial elementary embedding $j:V_{\lambda+2}\to V_{\lambda+2}$ such that $V_\kappa\prec V_\lambda\prec V_\gamma$ (for some $\gamma > \lambda > \kappa$).<cite>Baaz2011:Kurt</cite>:(definition 20.6, p. 455)
  
 
A '''Reinhardt cardinal''' is the critical point of a nontrivial elementary embedding $j:V\to V$ of the set-theoretic universe to itself.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
 
A '''Reinhardt cardinal''' is the critical point of a nontrivial elementary embedding $j:V\to V$ of the set-theoretic universe to itself.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
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A '''super Reinhardt''' cardinal $\kappa$, is a cardinal which is the critical point of elementary embeddings $j:V\to V$, with $j(\kappa)$ as large as desired.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
 
A '''super Reinhardt''' cardinal $\kappa$, is a cardinal which is the critical point of elementary embeddings $j:V\to V$, with $j(\kappa)$ as large as desired.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
  
For a proper class $A$, cardinal $κ$ is called '''$A$-super Reinhardt''' if for all ordinals $λ$ there is a non-trivial elementary embedding $j : V V$ such that $\mathrm{crit}(j) = κ$, $j(κ) > λ$ and $j(A) = A$. (where $j(A) := ⋃_{α∈\mathrm{OR}} j(A ∩ V_α)$)<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
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For a proper class $A$, cardinal $\kappa$ is called '''$A$-super Reinhardt''' if for all ordinals $\lambda$ there is a non-trivial elementary embedding $j : V \rightarrow V$ such that $\mathrm{crit}(j) = \lambda$, $j(\kappa)\gt\lambda$ and $j^+(A)=A$. (where $j^+(A) := \cup_{α∈\mathrm{Ord}} j(A ∩ V_α)$)<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
  
A '''totally Reinhardt''' cardinal is a cardinal $κ$ such that for each $A ∈ V_{κ+1}$, $\langle V_κ , V_{κ+1} \rangle \models \mathrm{ZF}_2 + \text{“There is an $A$-super Reinhardt cardinal”}$.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
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A '''totally Reinhardt''' cardinal is a cardinal $\kappa$ such that for each $A ∈ V_{κ+1}$, $(V_\kappa, V_{\kappa+1})\vDash \mathrm{ZF}_2 + \text{“There is an $A$-super Reinhardt cardinal”}$.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
  
== Relations ==
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Totally Reinhardt cardinals are the ultimate conclusion of the Vopěnka hierarchy. A cardinal is $n-$fold Vopěnka  if and only if, for every $A\subseteq V_\kappa$, there is some $\alpha\lt\kappa$ $\eta-$fold extendible for $A$ for every \(\eta\lt\kappa\), in that the witnessing embeddings fix $A\cap V_{\zeta_i}$. In its original conception Reinhardt cardinals were thought of as ultimate extendible cardinals, because if $j: V\rightarrow V$ is elementary, then so is $j\restriction V_{\kappa+\eta}: V_{\kappa+\eta}\rightarrow V_{j(\kappa+\eta)}$. It is as if one embedding works for all $\eta$.
* $\mathrm{WRA}$ (1) implies thet there are arbitrary large $\mathrm{I}_1$ and super $n$-huge cardinals. Kunen inconsistency does not apply to it. It is not known to imply $\mathrm{I}_0$.<cite>Corazza2010:TheAxiomOfInfinityAndJVV</cite>
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* $\mathrm{WRA}$ (1) does not need $j$ in the language. It hovewer requires another extension to the language of $\mathrm{ZFC}$, because otherwise there would be no weakly Reinhardt cardinals in $V$ because there are no weakly Reinhardt cardinals in $V_\kappa$ (if $\kappa$ is the least weakly Reinhardt) — obvious contradiction.<cite>Corazza2010:TheAxiomOfInfinityAndJVV</cite>
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==Relations==
* $\mathrm{WR}(\kappa)$ (1) implies that $\kappa$ is a measurable limit of [[supercompact]] cardinals and therefore is [[strongly compact]]. It is not known whether $\kappa$ must be supercompact itself. Requiring it to be [[extendible]] makes the theory stronger.<cite>Corazza2010:TheAxiomOfInfinityAndJVV</cite>
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$\mathrm{WRA}$ (1) implies thet there are arbitrary large $I1$ and super $n$-huge cardinals. Kunen inconsistency does not apply to it. It is not known to imply $I0$.<cite>Corazza2010:TheAxiomOfInfinityAndJVV</cite>
* Weakly Reinhardt cardinal(2) is inconsistent with $\mathrm{ZFC}$. $\mathrm{ZF} + \text{“There is a weakly Reinhardt cardinal(2)”} \implies \mathrm{Con}(\mathrm{ZFC} + \text{“There is a proper class of $\omega$-huge cardinals”})$ (at least here $\omega$-huge=I1) (Woodin, 2009).<cite>Baaz2011:Kurt</cite>
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* If $κ$ is super Reinhardt, then there exists $γ < κ$ such that $\langle V_γ , V_{γ+1} \rangle \models \mathrm{ZF}_2 + \text{“There is a Reinhardt cardinal”}$.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
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$\mathrm{WRA}$ (1) does not need $j$ in the language. It however requires another extension to the language of $\mathrm{ZFC}$, because otherwise there would be no weakly Reinhardt cardinals in $V$ because there are no weakly Reinhardt cardinals in $V_\kappa$ (if $\kappa$ is the least weakly Reinhardt) — obvious contradiction.<cite>Corazza2010:TheAxiomOfInfinityAndJVV</cite>
* Totally Reinhardt cardinals are obviously stronger than super Reinhardt.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
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* If $δ_0$ is the least [[Berkeley]] cardinal, then there is $γ < δ_0$ s.t. $\langle V_γ , V_{γ+1} \rangle \models \mathrm{ZF}_2 + \text{“There is a Reinhardt cardinal witnessed by $j$ and an ω-huge above $κ_ω(j)”$}$.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
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$\mathrm{WR}(\kappa)$ (1) implies that $\kappa$ is a measurable limit of [[supercompact]] cardinals and therefore is [[strongly compact]]. It is not known whether $\kappa$ must be supercompact itself. Requiring it to be [[extendible]] makes the theory stronger.<cite>Corazza2010:TheAxiomOfInfinityAndJVV</cite>
* Each club Berkeley cardinal is totally Reinhardt.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
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Weakly Reinhardt cardinal(2) is inconsistent with $\mathrm{ZFC}$. $\mathrm{ZF} + \text{“There is a weakly Reinhardt cardinal(2)”}\rightarrow\mathrm{Con}(\mathrm{ZFC} + \text{“There is a proper class of $\omega$-huge cardinals”})$ (At least here $\omega$-huge=$I1$) (Woodin, 2009). You can get this by seeing that $V_\gamma\vDash\forall\alpha\lt\lambda(\exists\kappa'\gt\alpha(I3(\kappa')\land\kappa'\lt\lambda))$.
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 +
If $\kappa$ is super Reinhardt, then there exists $\gamma\lt\kappa$ such that $(V_\gamma , V_{\gamma+1})\vDash \mathrm{ZF}_2 + \text{“There is a Reinhardt cardinal”}$.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
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 +
If $\delta_0$ is the least [[Berkeley]] cardinal, then there is $\gamma\lt\delta_0$ such that $(V_\gamma , V_{\gamma+1})\vDash\mathrm{ZF}_2+\text{“There is a Reinhardt cardinal witnessed by $j$ and an $\omega$-huge above $\kappa_\omega(j)”$}$. (Here $\omega-$huge means $I3$). <cite>Bagaria2017:LargeCardinalsBeyondChoice</cite> Each club Berkeley cardinal is totally Reinhardt.<cite>Bagaria2017:LargeCardinalsBeyondChoice</cite>
  
 
{{References}}
 
{{References}}

Revision as of 09:20, 21 August 2019

The existence of Reinhardt cardinals has been refuted in $\text{ZFC}_2$ and $\text{GBC}$ by Kunen (Kunen inconsistency), the term is used in the $\text{ZF}_2$ context, although some mathematicians suspect that they are inconsistent even there.

Definitions

A weakly Reinhardt cardinal(1) is the critical point $\kappa$ of a nontrivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$ such that $V_\kappa\prec V$ ($\mathrm{WR}(\kappa)$. Existence of $\kappa$ is Weak Reinhardt Axiom ($\mathrm{WRA}$) by Woodin).[1]:p.58

A weakly Reinhardt cardinal(2) is the critical point $\kappa$ of a nontrivial elementary embedding $j:V_{\lambda+2}\to V_{\lambda+2}$ such that $V_\kappa\prec V_\lambda\prec V_\gamma$ (for some $\gamma > \lambda > \kappa$).[2]:(definition 20.6, p. 455)

A Reinhardt cardinal is the critical point of a nontrivial elementary embedding $j:V\to V$ of the set-theoretic universe to itself.[3]

A super Reinhardt cardinal $\kappa$, is a cardinal which is the critical point of elementary embeddings $j:V\to V$, with $j(\kappa)$ as large as desired.[3]

For a proper class $A$, cardinal $\kappa$ is called $A$-super Reinhardt if for all ordinals $\lambda$ there is a non-trivial elementary embedding $j : V \rightarrow V$ such that $\mathrm{crit}(j) = \lambda$, $j(\kappa)\gt\lambda$ and $j^+(A)=A$. (where $j^+(A) := \cup_{α∈\mathrm{Ord}} j(A ∩ V_α)$)[3]

A totally Reinhardt cardinal is a cardinal $\kappa$ such that for each $A ∈ V_{κ+1}$, $(V_\kappa, V_{\kappa+1})\vDash \mathrm{ZF}_2 + \text{“There is an $A$-super Reinhardt cardinal”}$.[3]

Totally Reinhardt cardinals are the ultimate conclusion of the Vopěnka hierarchy. A cardinal is $n-$fold Vopěnka if and only if, for every $A\subseteq V_\kappa$, there is some $\alpha\lt\kappa$ $\eta-$fold extendible for $A$ for every \(\eta\lt\kappa\), in that the witnessing embeddings fix $A\cap V_{\zeta_i}$. In its original conception Reinhardt cardinals were thought of as ultimate extendible cardinals, because if $j: V\rightarrow V$ is elementary, then so is $j\restriction V_{\kappa+\eta}: V_{\kappa+\eta}\rightarrow V_{j(\kappa+\eta)}$. It is as if one embedding works for all $\eta$.

Relations

$\mathrm{WRA}$ (1) implies thet there are arbitrary large $I1$ and super $n$-huge cardinals. Kunen inconsistency does not apply to it. It is not known to imply $I0$.[1]

$\mathrm{WRA}$ (1) does not need $j$ in the language. It however requires another extension to the language of $\mathrm{ZFC}$, because otherwise there would be no weakly Reinhardt cardinals in $V$ because there are no weakly Reinhardt cardinals in $V_\kappa$ (if $\kappa$ is the least weakly Reinhardt) — obvious contradiction.[1]

$\mathrm{WR}(\kappa)$ (1) implies that $\kappa$ is a measurable limit of supercompact cardinals and therefore is strongly compact. It is not known whether $\kappa$ must be supercompact itself. Requiring it to be extendible makes the theory stronger.[1]

Weakly Reinhardt cardinal(2) is inconsistent with $\mathrm{ZFC}$. $\mathrm{ZF} + \text{“There is a weakly Reinhardt cardinal(2)”}\rightarrow\mathrm{Con}(\mathrm{ZFC} + \text{“There is a proper class of $\omega$-huge cardinals”})$ (At least here $\omega$-huge=$I1$) (Woodin, 2009). You can get this by seeing that $V_\gamma\vDash\forall\alpha\lt\lambda(\exists\kappa'\gt\alpha(I3(\kappa')\land\kappa'\lt\lambda))$.

If $\kappa$ is super Reinhardt, then there exists $\gamma\lt\kappa$ such that $(V_\gamma , V_{\gamma+1})\vDash \mathrm{ZF}_2 + \text{“There is a Reinhardt cardinal”}$.[3]

If $\delta_0$ is the least Berkeley cardinal, then there is $\gamma\lt\delta_0$ such that $(V_\gamma , V_{\gamma+1})\vDash\mathrm{ZF}_2+\text{“There is a Reinhardt cardinal witnessed by $j$ and an $\omega$-huge above $\kappa_\omega(j)”$}$. (Here $\omega-$huge means $I3$). [3] Each club Berkeley cardinal is totally Reinhardt.[3]

References

  1. Corazza, Paul. The Axiom of Infinity and transformations $j: V \to V$. Bulletin of Symbolic Logic 16(1):37--84, 2010. www   DOI   bibtex
  2. Baaz, M and Papadimitriou, CH and Putnam, HW and Scott, DS and Harper, CL. Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press, 2011. www   bibtex
  3. Bagaria, Joan. Large Cardinals beyond Choice. , 2017. www   bibtex
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