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Joel David Hamkins
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If you have the semi-ring structure, with $+$ etc., then every other nonstandard element is even and every other element is odd, so the types within a $Z$ chain are not identical.

If you just have the order, then the types are identical, since there are numerous automorphisms, translating within a $Z$-chain or moving various $Z$ chains.

Joel David Hamkins
  • 236.3k
  • 44
  • 777
  • 1.4k