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    Non-standard real analysis can prove results in real anal... — Carmelics
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    Non-standard real analysis can prove results in real analysis that were not first proven by standard real analysis methods.

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    • There are cases of results in real analysis that were first proven using non-standard real analysis.
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    Reasons Against

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    Reason against 1 of 2
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    • 1.Historical priority of discovery does not establish logical or mathematical independence of the method used.
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    • 2.Robinson's 1966 non-standard proofs of results like the Bernstein-Robinson theorem were rapidly given standard reformulations by Halmos and others.
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    • 3.A result 'first proven' by non-standard means but immediately translatable into standard means reveals a sociological fact, not a mathematical one about the claim's provability.
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    Reason against 2 of 2
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    • 1.Any theorem provable via non-standard analysis is provable via standard analysis by the transfer principle (Łoś's theorem).
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    • 2.If standard proofs always exist for every non-standard result, non-standard methods yield no genuinely novel theorems, only novel proofs.
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    A result 'first proven' by non-standard means but immediately translatable into ...Any theorem provable via non-standard analysis is provable via standard analysis...Historical priority of discovery does not establish logical or mathematical inde...If standard proofs always exist for every non-standard result, non-standard meth...
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    Robinson's 1966 non-standard proofs of results like the Bernstein-Robinson theor...There are cases of results in real analysis that were first proven using non-sta...

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    There are cases of results in real analysis that were first proven usi...93%Non-standard analysis provides simpler and more intuitive proofs of ma...89%The approaches pioneered by Robinson and Nelson yield proofs of theore...85%These proofs are, in some sense, simpler than those produced by standa...82%

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    The approaches pioneered by Robinson and Nelson do not allow us to prove results about the standard real numbers that cannot be proved using standard real analysis. However, these approaches do provide simpler—and, in some sense, more intuitive—proofs of many theorems of standard real analysis. (On the pedagogical benefits of non-standard analysis, see, for example, Keisler (1976)). And there are cases of results in real analysis that were first proven using non-standard real analysis (see, for
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    claim
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    3 (1 for, 2 against)
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