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First Incompleteness Theorem, according to WIKIpedia: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F."

This is disturbing. Can I instead say this more exact version: "Any consistent formal system F, which contains the Peano axioms, is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F."

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    $\begingroup$ It would be more accurate to replace "Peano axioms" by "Robinson axioms". $\endgroup$ – Asaf Karagila Jan 23 '18 at 19:39
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    $\begingroup$ Even more, it is enough to talk of interpretability rather than containment. Also, the "formal system" should be a recursively enumerable (axiomatization of) a first-order theory. And if you want to be yet more exact, you should say something about the meta-theory in which this result is being established. $\endgroup$ – Andrés E. Caicedo Jan 23 '18 at 19:52
  • $\begingroup$ Aside: you need additional hypotheses on $F$; e.g. being generated by a set of axioms that is recursively enumerable. $\endgroup$ – Hurkyl Jan 23 '18 at 19:53
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    $\begingroup$ @AsafKaragila Are the Robinson axioms minimal, in the sense that any theory that has the incompleteness property can do Robinson arithmetic? $\endgroup$ – eyeballfrog Jan 23 '18 at 19:55
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    $\begingroup$ Hold on! The very Wikipedia page you (the OP) are quoting has a whole section already explaining what it takes to enable "a certain amount of elementary arithmetic [to be] carried out". Perhaps reread the whole page more carefully?? $\endgroup$ – Peter Smith Jan 23 '18 at 20:18

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