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  "body": "\n## Summary\n\nAn elementary embedding is a map between two [[structures|983GZBQYT4]] of the same signature that preserves and reflects the truth of every first-order formula.\nIts image is an [[elementary substructure|8WZF9SBP39]] of the target, isomorphic to the source.\nElementary embeddings are the natural maps of model theory, and in set theory they are used to define large cardinals.\n\n## Explanations\n\n### L1 human\n\nAn ordinary embedding copies a structure into another one while keeping the operations and relations.\nAn elementary embedding keeps much more: every first-order statement about elements holds before the map exactly when it holds after.\nThe inclusion of $(\\Q, \u003c)$ into $(\\R, \u003c)$ is elementary.\nThe map $n \\mapsto n + 1$ from $(\\N, \u003c)$ to $(\\N, \u003c)$ is an embedding, but not an elementary one: $0$ has no element below it, while its image $1$ has $0$ below it.\n",
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