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  "body": "\n## Summary\n\nIn mathematical logic, more specifically in the proof theory of [[first-order theories|086QKPF0A6]], an **extension by definition** formalizes the introduction of a new symbol by means of a definition. For example, it is common in naive set theory to introduce a symbol $\\emptyset$ for the [[set|0BSJ7RQ9BW]] that has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant $\\emptyset$ and the new axiom $\\forall x(x\\notin\\emptyset)$, meaning \"for all *x*, *x* is not a member of $\\emptyset$\". It can then be proved that doing so adds essentially nothing to the old theory, as should be expected from a definition. More precisely, the new theory is a conservative extension of the old one.\n",
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