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Todd Eisworth
Todd Eisworth
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The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets withthewith the property that for any $A\subseteq\omega_1$, there is a $\delta<\omega_1$ (equivalently, stationarily many $\delta<\omega_1$) for which $A\cap\delta= A_\delta$.

$\diamondsuit$ holds in the constructible universe $L$, and implies the Continuum Hypothesis. The axiom is also a key ingredient in many constructions, and research over the past half-century has given us a good understanding of when the use of $\diamondsuit$ is necessary, in the sense that it cannot be replaced by simply assuming the Continuum Hypothesis.

There are many strengthenings of $\diamondsuit$ that have been studied and utilized in constructions, e.g. $\diamondsuit^*$, $\diamondsuit^+$, $\diamondsuit^\sharp$, and of course $V=L$).

My question is: what are some examples of statements $\Phi$ where a strengthening of $\diamondsuit$ is known to imply $\Phi$, but it is still unknown if $\diamondsuit$ itself implies $\Phi$? How fertile is this terrain?

The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets withthe property that for any $A\subseteq\omega_1$, there is a $\delta<\omega_1$ (equivalently, stationarily many $\delta<\omega_1$) for which $A\cap\delta= A_\delta$.

$\diamondsuit$ holds in the constructible universe $L$, and implies the Continuum Hypothesis. The axiom is also a key ingredient in many constructions, and research over the past half-century has given us a good understanding of when the use of $\diamondsuit$ is necessary, in the sense that it cannot be replaced by simply assuming the Continuum Hypothesis.

There are many strengthenings of $\diamondsuit$ that have been studied and utilized in constructions, e.g. $\diamondsuit^*$, $\diamondsuit^+$, $\diamondsuit^\sharp$, and of course $V=L$).

My question is: what are some examples of statements $\Phi$ where a strengthening of $\diamondsuit$ is known to imply $\Phi$, but it is still unknown if $\diamondsuit$ itself implies $\Phi$? How fertile is this terrain?

The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets with the property that for any $A\subseteq\omega_1$, there is a $\delta<\omega_1$ (equivalently, stationarily many $\delta<\omega_1$) for which $A\cap\delta= A_\delta$.

$\diamondsuit$ holds in the constructible universe $L$, and implies the Continuum Hypothesis. The axiom is also a key ingredient in many constructions, and research over the past half-century has given us a good understanding of when the use of $\diamondsuit$ is necessary, in the sense that it cannot be replaced by simply assuming the Continuum Hypothesis.

There are many strengthenings of $\diamondsuit$ that have been studied and utilized in constructions, e.g. $\diamondsuit^*$, $\diamondsuit^+$, $\diamondsuit^\sharp$, and of course $V=L$).

My question is: what are some examples of statements $\Phi$ where a strengthening of $\diamondsuit$ is known to imply $\Phi$, but it is still unknown if $\diamondsuit$ itself implies $\Phi$? How fertile is this terrain?

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Todd Eisworth
Todd Eisworth
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On the role of $\diamondsuit$

The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets withthe property that for any $A\subseteq\omega_1$, there is a $\delta<\omega_1$ (equivalently, stationarily many $\delta<\omega_1$) for which $A\cap\delta= A_\delta$.

$\diamondsuit$ holds in the constructible universe $L$, and implies the Continuum Hypothesis. The axiom is also a key ingredient in many constructions, and research over the past half-century has given us a good understanding of when the use of $\diamondsuit$ is necessary, in the sense that it cannot be replaced by simply assuming the Continuum Hypothesis.

There are many strengthenings of $\diamondsuit$ that have been studied and utilized in constructions, e.g. $\diamondsuit^*$, $\diamondsuit^+$, $\diamondsuit^\sharp$, and of course $V=L$).

My question is: what are some examples of statements $\Phi$ where a strengthening of $\diamondsuit$ is known to imply $\Phi$, but it is still unknown if $\diamondsuit$ itself implies $\Phi$? How fertile is this terrain?