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Definition: An ordered set is order-complete if any nonempty subset with an upper bound, has a lowest upper bound or supremo.

Notation: We denote the system of first-order Peano Axioms (along with axioms for addition and multiplication) by PA1.

1.- Can we express the order-completeness of $\mathbb{N}$ using first-order logic? How does it look?

2.- Can we prove that natural numbers are order-complete using PA1 or it has to be considered as an axiom?

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    $\begingroup$ I answered on math.stackexchange. $\endgroup$ Commented Feb 27, 2014 at 4:01
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    $\begingroup$ This question appears to be off-topic because it has been answered elsewhere. $\endgroup$
    – Yemon Choi
    Commented Feb 27, 2014 at 4:56

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