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  title: 'Peano Arithmetic',
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  text: 'Peano Arithmetic is a particular set of axioms and rules which allow you to prove theorems about the natural numbers.\n\nThese rules were formulated by the Italian mathematician [Giuseppe Peano](https://en.wikipedia.org/wiki/Giuseppe_Peano) in 1889. They can be expressed as follows:\n\nLet our language consist of the symbols $\\left\\{(,),\\wedge,\\vee,\\neg,\\to,\\leftrightarrow,\\in,\\forall,\\exists,=,+,\\cdot,O,S,N \\right\\}$ and an infinite set of variable symbols, which we will denote as $x, y, z, \\dots$ (since three symbols is usually enough to denote infinitely many symbols). \n\nWe would like to interpret these symbols as representing our intuitive notions of logical and arithmetical operators, interpreting $O$ as the number 0, $S$ as the successor operation (thus $SO$ represents 1, $SSO$ represents 2, etc), and $N$ as the set of natural numbers.\n\nWe would furthermore like to create some formal rules such that we can derive certain true statements of arithmetic, like $SO+SO=SSO$ or $\\forall x \\in N\\; Sx \\cdot Sx = x\\cdot x + SSO \\cdot x + SO$, but not derive false statements like $\\exists x\\in N \\; SSO\\cdot x = SSSO$.',
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