The hyperreals and standard reals satisfy the transfer pr... — Carmelics

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The hyperreals and standard reals satisfy the transfer principle for first-order logical results, but behave differently for results about sets.

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Reasons For

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Reason for

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1.For results stated in a first-order logical language, the hyperreals and the standard reals satisfy the transfer principle.
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2.Every bounded set of standard reals has a least upper bound.
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3.The set of infinitesimal hyperreals is bounded but has no least upper bound.
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Reasons Against

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Reason against 1 of 2

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1.The set of infinitesimal hyperreals lacks a least upper bound because 'least upper bound' in NSA refers to an internal set, not an external one.
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2.The completeness axiom transfers correctly to hyperreals when restricted to internal sets, making the supporting argument's P3 a category error.
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3.Robinson's transfer principle was explicitly formulated to apply only to internal properties, so citing external sets as counterexamples misrepresents the theorem's scope.
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