Are Induction and Well-Ordering Equivalent?

We may also note that the axiomatic system consisting of axioms (1)-(5) admits only models that are isomorphic to the natural numbers, and since the natural numbers are well-ordered, in this system well-ordering ( To the human mind, the natural numbers may be a more primitive intuitive concept than sets, but that notwithstanding, the more common reduction in texts on the foundations of mathematics is usually that the concept of natural number is reduced to the concept of sets. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="double-struck">Q</mi></math> Note that the natural numbers were introduced by means of Peano's axioms, so if one removes the principle of induction from the axiom set and replaces it with the well-ordering property, the proof of Theorem 1 is incorrect. Bachelor's thesis in mathematics, Dept. of Mathematics and Mathematical Statistics, Umeå University, 2016. [Extracted from the article]