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Assuming well ordering principle one can prove Archimedean property for the set of positive Integers (Burton). Is the converse also true? That is assuming Archimedean Property, can we establish well ordering property for the set of positive integers? Actually I want to know whether "Archimedean Property" is equivalent to "Well Ordering Principle" over the set of positive integers ?

Kaustubh Dutta
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    The positive rationals have the Archimedean Property, but not the well-ordering property, right? – Gerry Myerson May 01 '21 at 12:49
  • That's true. I have asked the question in the context of positive integers. – Kaustubh Dutta May 01 '21 at 12:56
  • And what exsctly is thst context? What axions are you assuming? – Gerry Myerson May 01 '21 at 13:20
  • In Topology by Munkres, set of positive integers is defined as smallest inductive set of real numbers (page 32) and using that fact well ordering principle is established as a theorem [ Theorem 4.1, Topology, Munkres]. Asuuming this definition of positive integer, can we establish equivalence of well ordering principle and Archimedean property for positive integers? Using proof by contradiction one can easily establish Archimedean Property for integers from the well ordering principle. I am curious about the converse part. – Kaustubh Dutta May 01 '21 at 14:10
  • If you are assuming the positive integers are the smallest inductive set of reals, then (as you say) you can deduce well-ordering without appealing to the Archimedean property, so you can certainly deduce well-ordering from the Archimedean property in a trivial way, by just including the phrase, "by the way, the integers satisfy the Archimedean property" in the middle of your proof of well-ordering. I don't think that's what you want, but then it's still not clear what you do want. – Gerry Myerson May 01 '21 at 22:43

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