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If we present the axioms of the Real numbers in second order logic (SoL) then we need the completeness axiom to uniquely determine the structure. Otherwise, without the completeness axiom, we cannot conclude that all the models of this second order theory are the same up to an isomorphism.

Similarly, in Peano axioms, if we remove the induction axiom (the 9th axiom), then we might end up with sets like $\{1,2,3,\cdots\}\cup \{a,b\}$, which satisfy all the other axioms but are not the same as the set of Natural numbers. So it is this induction axiom that allows for unique determination of set of Natural numbers using the SoL axioms.

Hence, one might argue that both these axioms are serving the same purpose, in particular, uniquely determining the corresponding structures using SoL axioms.

Here are the questions:

  1. Is the above argument correct?
  2. In SoL, can a second order axiom (s) be added to a set of axioms to allow for unique determination of underlying structure?
  3. What would be the characteristics of the axioms needed to uniquely determine a given structure?
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    $\begingroup$ I don't understand the votes to close - especially the one citing this as "not about mathematics," that's absolutely ridiculous. $\endgroup$ May 24, 2017 at 21:51

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