Square principles

Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial.

Definition. A square principle for a cardinal $\theta$ asserts the existence of a sequence $\Gamma=\langle C_\alpha \mid \alpha<\theta\rangle$ such that for all $\alpha<\theta$, $C_\alpha$ is a closed subset of $\theta$.

Characteristics

  1. $S(\Gamma)$, the support of $\Gamma$, is the set $\{\alpha<\theta\mid C_\alpha\subseteq\sup(C_\alpha)=\alpha\}$;
  2. $w(\Gamma)$, the width of $\Gamma$, is the cardinal $\sup\{ |\mathcal T(\Gamma,\alpha)|\mid \alpha<\theta\}$, where $$\mathcal T(\Gamma,\alpha)=\{ C_\beta\cap\alpha\mid \beta<\theta,\sup(C_\beta\cap\alpha)=\alpha\};$$
  3. Nontriviality condition.

Examples:

(Jensen, 1972). In $L$, for every infinite cardinal $\kappa$, $\square_\kappa$ holds. That is, there exists a sequence $\Gamma=\langle C_\alpha\mid \alpha<\kappa^+\rangle$ with

  1. $S(\Gamma)=\{\alpha<\kappa^+\mid \alpha\text{ limit}\}$;
  2. $w(\Gamma)=1$;
  3. $\text{otp}(C_\alpha)\le\kappa$ for all $\alpha<\kappa^+$.

(Jensen, 1972). In $L$, if $\kappa$ is a regular uncountable cardinal that is not weakly compact, then there exists a sequence $\Gamma=\langle C_\alpha\mid \alpha<\kappa\rangle$ with

  1. $S(\Gamma)=\{\alpha<\kappa\mid \alpha\text{ limit}\}$;
  2. $w(\Gamma)=1$;
  3. there exists a stationary subset $E\subset\kappa$ such that $E$ contains no accumulation points of $C_\alpha$ for all $\alpha<\kappa$.

(Todorcevic, 1984). PFA refutes the existence of a sequence $\Gamma=\langle C_\alpha\mid \alpha<\theta\rangle$ with

  1. $S(\Gamma)=\{\alpha<\theta\mid \text{cf}(\alpha)=\omega_1\}$, $\theta=\text{cf}(\theta)>\omega_1$;
  2. $w(\Gamma)=1$;
  3. there exists no club $C$ in $\theta$ such that $C\cap\alpha=C_\alpha$ for every $\alpha\in S(\Gamma)$ which is an accumulation point of $C$.

(Baumgartner, 198?, Cummings-Magidor, 2011, Brodsky-Rinot, 2016). PFA is consistent with $\boxminus_{\kappa,\ge\aleph_2}$ for every singular cardinal $\kappa$, where $\boxminus_{\kappa,\ge\chi}$ asserts the existence of a sequence $\Gamma=\langle C_\alpha\mid \alpha<\kappa^+\rangle$ with

  1. $S(\Gamma)\supseteq\{\alpha<\kappa^+\mid \text{cf}(\alpha)\ge\chi\}$;
  2. $w(\Gamma)=1$;
  3. $\text{otp}(C_\alpha)\le\kappa$ for all $\alpha<\kappa^+$.

(Velickovic, 1986, Todorcevic, 1987, Shelah-Stanley, 1988). If $\theta=\text{cf}(\theta)>\omega_1$ is not weakly compact in L, then $\square(\theta)$ holds. That is, there exists a sequence $\Gamma=\langle C_\alpha  \mid \alpha<\theta\rangle$ with

  1. $S(\Gamma)=\{\alpha<\theta\mid \alpha\text{ limit}\}$;
  2. $w(\Gamma)=1$;
  3. there exists no club $C$ in $\theta$ such that $C\cap\alpha=C_\alpha$ for every $\alpha\in S(\Gamma)$ which is an accumulation point of $C$.

(the proof in Velickovic, 1986 focuses on the construction of $C_\alpha$ for $\alpha$ which is a regular cardinal. The proof in Todorcevic, 1987 is more detailed, and the proof in Shelah-Stanley, 1988 is the most detailed.)

(Rinot, 2016). If GCH holds and $\theta>\omega_1$ is a successor cardinal which is not weakly compact in L, then $\boxtimes^-(\theta)$ holds. That is, there exists a sequence $\Gamma=\langle C_\alpha  \mid \alpha<\theta\rangle$ with

  1. $S(\Gamma)=\{\alpha<\theta\mid \alpha\text{ limit}\}$;
  2. $w(\Gamma)=1$;
  3. for every cofinal subset $A\subseteq\theta$, there exists some nonzero limit ordinal $\alpha<\kappa$ such that $\sup(\text{nacc}(C_\alpha)\cap A)=\alpha$.

The fact that $\boxtimes^-(\theta)$ implies $\square(\theta)$ may be found in here.

(Shelah, 1991. See here). If $\lambda<\kappa$ are regular cardinals, and $S$ is a stationary subset of $E^{\kappa^+}_\lambda$, then there exists a sequence $\Gamma=\langle C_\alpha\mid \alpha<\kappa^+\rangle$ with

  1. $S(\Gamma)\cap S$ is stationary;
  2. $w(\Gamma)=1$;
  3. $\text{otp}(C_\alpha)\le\lambda$ for all $\alpha<\kappa^+$.

(Schimmerling, 1995). $\square_{\kappa,\lambda}$ asserts the existence of a sequence $\Gamma=\langle C_\alpha  \mid \alpha<\kappa^+\rangle$ with

  1. $S(\Gamma)=\{\alpha<\kappa^+\mid \alpha\text{ limit}\}$;
  2. $w(\Gamma)\le\lambda$;
  3. $\text{otp}(C_\alpha)\le\kappa$ for all $\alpha<\kappa^+$.

(Krueger, 2013, extending Jensen, 1972). Weak square on $\kappa$ asserts the existence of a sequence $\Gamma=\langle C_\alpha  \mid \alpha<\kappa\rangle$ with

  1. $S(\Gamma)$ is a club in $\kappa$ consisting of singular ordinals (so $\kappa$ is non-Mahlo);
  2. $|\mathcal T(\Gamma,\alpha)|<\kappa$ for all $\alpha<\kappa$;
  3. $\text{otp}(C_\alpha)<\alpha$ for all $\alpha<\kappa$.

(Todorcevic and Torres Perez, 2014). Rado’s conjecture refutes the existence of a sequence $\Gamma=\langle C_\alpha  \mid \alpha<\theta\rangle$ with

  1. $S(\Gamma)=\{\alpha<\theta\mid \alpha\text{ limit}\}$, $\theta=\text{cf}(\theta)>\omega_1$, not the successor of a singular cardinal of countably cofinality;
  2. $w(\Gamma)\le\aleph_0$;
  3. there exists no sequence $\langle A_\alpha\mid \alpha<\theta\rangle$ of countable subsets of $\theta$ satisfying the following. For all $\alpha<\beta<\theta$, there exist $\tau\in S(\Gamma)$ and $\gamma<\delta$ with $(\gamma,\delta)\in A_\alpha\times A_\beta$ such that $\sup(C_\tau\cap\gamma)=\gamma$, $\sup(C_\tau\cap\delta)=\delta$.

(Neeman, 2014). $\square_{\kappa,\lambda}^{\text{ta}}$ asserts the existence of a sequence $\Gamma=\langle C_\alpha \mid \alpha<\kappa^+\rangle$ with

  1. $S(\Gamma)=\{\alpha<\kappa^+\mid \alpha\text{ limit}\}$;
  2. $w(\Gamma)\le\lambda$;
  3. $\text{otp}(C_\alpha)<\kappa$ for all $\alpha<\kappa^+$ of cofinality $<\kappa$, and for all $C,D\in\mathcal T(\Gamma,\alpha)$, there exists $\beta<\alpha$ with $C\setminus\beta=D\setminus\beta$.

 

For more square principles, see the papers by Cummings and Schimmerling, Krueger and Schimmerling, and Brodsky and Rinot. The following presentation by Hiroshi Sakai presents the state-of-the-art of the study of failure of these principles.
Foreman and Magidor wrote an interesting paper on a couple of weak forms of square, but these principles fit more into the so-called approachability principles. What is the difference between square principles and approachability principles? the major ones are that the sets $C_\alpha$ do not have to be closed, and that instead of measuring the width of $\Gamma$, one seeks to occur into (some or all) bounded subsets of $C_\alpha$ sometime before arriving to $\alpha$.
An important paper on approachability principles is the one by Dzamonja and Shelah. Finally, it may worth metnioning that in Section 4 of this paper, I am constructing a model where (weak) square fails at $\aleph_\omega$, yet a (quite useful) approachability principle holds.

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13 Responses to Square principles

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  5. Ur Yaar says:

    Thanks for this list! Shouldn’t there also be a coherence condition? At least in Jensen’s original square we have that at any limit point $\alpha$ of $C_\beta$, we have $C_\beta
    \cap\alpha = C_\alpha$ and I don’t see it implied by the other conditions.

    • saf says:

      Hi Ur!
      Isn’t this the same as saying that the width is 1?

      • Ur Yaar says:

        Oh OK, I thought that taking a sup isn’t enough since I thought you might get places where $\mathscr{T}$ is empty but now I see that when you take $\alpha=\beta$ for $\alpha$ in the support you get at least one element, and the others must be the same. Thanks!

  6. Ur Yaar says:

    I’m interested in knowing whether there could be (in L) a “Reflecting” square sequence – a sequence such that some of its initial segments are also square sequences in the respective lower cardinals. I gave an exact formulation in MathOverflow:
    https://mathoverflow.net/questions/426249/reflecting-square-sequences
    Do you know if it is possible?

    It resembles the result by Shelah (1991) you present here, but I don’t want that _all_ elements in the sequence will have low order-type, but only the ones below some cardinals.

    • saf says:

      Have a look at Theorem 5.1 of Jensen’s “The fine structure of the constructible hierarchy” from 1972. He gets there a global square sequence on the class of all singular ordinals. See also: http://www.logic.univie.ac.at/~sdf/papers/joint.james.pdf

      • Ur Ya’ar says:

        Hmm yes I know of the global square, but dismissed it because I thought I do need something that includes regulars as well. But now that you mention it, maybe using clubs on singulars will suffice for my needs. I need to check. Thanks!

      • Ur Yaar says:

        OK this seems to be what I’m after indeed!
        Is there a name/notation for a sequence which is a bounded part of a global square sequence? I.e. a sequence
        $\mathcal{C}=\left\langle C_{\alpha}\mid\alpha<\theta \text{ a singular ordinal}\right\rangle $ for some ordinal $\theta$ satisfying
        1. $C_{\alpha}$ is club in $\alpha$;
        2. $otp(C_{\alpha})<\alpha$;
        3. If $\beta\in\mathrm{Lim}(C_{\alpha})$ then $C_{\beta}=C_{\alpha}\cap\beta$.

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