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  "body": "\n## Summary\n\nAn infinitary logic is an extension of [[first-order logic|086QKPF0A6]] that allows formulas of infinite length.\nExamples are conjunctions of infinitely many formulas and infinite blocks of quantifiers.\nThe logics $L_{\\kappa\\lambda}$ are the standard family, and $L_{\\omega_1\\omega}$, with countable conjunctions and finite quantifier blocks, is the most studied one.\nThese logics can express properties that first-order logic cannot, but they lose the compactness theorem.\n\n## Explanations\n\n### L1 human\n\nIn ordinary logic a formula is a finite string of symbols.\nInfinitary logic allows a formula to say \"$A_1$ or $A_2$ or $A_3$ or ...\" without end.\nFor example, \"every element $x$ of the group has some finite order\" is the infinite disjunction \"$x = e$ or $x^2 = e$ or $x^3 = e$ or ...\".\nNo single first-order sentence says this about all groups.\n",
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