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  "body": "\n## Definition\n\nLet $L$ be a signature and let $\\kappa \\ge \\lambda$ be infinite [[cardinals|0EBACQFAZW]].\nThe formulas of the infinitary logic $L_{\\kappa\\lambda}$ are generated by the following rules.\n\n1. Each atomic [[formula|HYJ10AF6RT]] of [[first-order logic|086QKPF0A6]] over $L$ is a formula.\n2. If $\\varphi$ is a formula, then $\\neg\\varphi$ is a formula.\n3. If $\\Phi$ is a set of formulas with $\\abs{\\Phi} \u003c \\kappa$ whose free variables together number fewer than $\\lambda$, then $\\bigwedge \\Phi$ and $\\bigvee \\Phi$ are formulas.\n4. If $\\varphi$ is a formula and $V$ is a set of variables with $\\abs{V} \u003c \\lambda$, then $\\forall V \\, \\varphi$ and $\\exists V \\, \\varphi$ are formulas.\n\nTruth in an $L$-[[structure|983GZBQYT4]] is defined as in first-order logic: $\\bigwedge \\Phi$ holds when every member of $\\Phi$ holds, and $\\bigvee \\Phi$ when some member holds.\n\n## Remarks\n\nThe logic $L_{\\omega\\omega}$ is first-order logic, since then all conjunctions, disjunctions and quantifier blocks are finite.\nThe logic $L_{\\omega_1\\omega}$ allows countable conjunctions and disjunctions and finite quantifier blocks.\n\nExample: in the signature of groups, the $L_{\\omega_1\\omega}$ sentence $\\forall x \\, \\bigvee_{n \\ge 1} (x^n = e)$ holds exactly in the torsion groups.\nHere $x^n$ is the term $x \\cdot x \\cdots x$ with $n$ factors.\n\nNon-example: the expression $\\forall x_1 \\, \\forall x_2 \\, \\forall x_3 \\cdots \\varphi$ with infinitely many quantified variables is not a formula of $L_{\\omega_1\\omega}$, because clause 4 allows only finite sets of variables when $\\lambda = \\omega$.\nIt is a formula of $L_{\\omega_1\\omega_1}$ when $\\varphi$ is.\n\nThe compactness theorem fails for $L_{\\omega_1\\omega}$, but the downward Löwenheim–Skolem theorem holds for its countable fragments.\n\n## Explanations\n\n### L1 human\n\nInfinitary logic is first-order logic with permission to write infinitely long \"and\" and \"or\".\nThe two cardinals bound how long these may be and how many variables a quantifier may bind at once.\n",
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