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  "body": "\n## Definition\n\nLet $\\omega_1$ be the least uncountable ordinal.\nThe diamond principle $\\diamondsuit$ is the following statement.\nThere is a [[sequence|0QE8STGJH2]] $(A_\\alpha)_{\\alpha \u003c \\omega_1}$ with $A_\\alpha \\subseteq \\alpha$ for each $\\alpha$, such that for each $A \\subseteq \\omega_1$ the set\n$$\\set{\\alpha \u003c \\omega_1 : A \\cap \\alpha = A_\\alpha}$$\nis [[stationary|6FDYJVG1CK]] in $\\omega_1$.\nSuch a sequence is called a $\\diamondsuit$-sequence.\n\n## Remarks\n\nExample: Jensen proved that $\\diamondsuit$ holds in the constructible universe $L$, so it is consistent with ZFC if ZFC is consistent.\nNon-example: the sequence with $A_\\alpha = \\varnothing$ for all $\\alpha$ is not a $\\diamondsuit$-sequence.\nFor $A = \\omega_1$, the set $\\set{\\alpha : \\alpha \\cap \\omega_1 = \\varnothing} = \\set{0}$ is not stationary.\n$\\diamondsuit$ implies the continuum hypothesis: each $A \\subseteq \\omega$ equals $A \\cap \\alpha = A_\\alpha$ for some $\\alpha \u003e \\omega$, so there are at most $\\aleph_1$ subsets of $\\omega$.\n$\\diamondsuit$ implies that there is a Suslin tree, and so it settles Suslin's problem negatively.\nMore generally, $\\diamondsuit_\\kappa(S)$ is defined for a regular uncountable cardinal $\\kappa$ and a stationary $S \\subseteq \\kappa$, with guesses only at points of $S$.\n",
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