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In The Continuum, Hermann Weyl writes the following (page 95 in the English translation):

The system which, for the moment, we shall call "hyperanalysis" arises if, starting from the level attained in §3 of this chapter, we lay a new foundation for pure number theory, a foundation in which we admit the real numbers as a new basic category alongside the naturals. ... This new system certainly does not coincide with our version of analysis. On the contrary, in hyperanalysis there are, e.g., more sets of real numbers than in analysis. For hyperanalysis admits sets in whose definition "there is" appears in connections with "a real number". Thus, hyperanalysis contains neither Cauchy's convergence principle nor, in general, our theorems about continuous functions.

My question is: What does he mean?

The following should help to clarify why I am confused: What does he refer to by "the real numbers" in the first sentence? Only the "level 1" real numbers that can be defined by arithmetical comprehension? Or all possible real numbers? In the first case, "hyperanalysis" would seem to mean analysis with level 2 real numbers, which I suppose would behave as nicely as his default kind of analysis with only level 1 real numbers, so that the italized sentence is false. In the latter case, there would be circularity in hyperanalysis, and then it is odd that Weyl does not point that out but only makes the weaker point about Cauchy's convergence principle and theorems about continuous functions.

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Weyl's system is predicative; in it, for example, the well-known Least Upper Bound principle is not derivable.

But Weyl is able to derive the Least Upper Bound principle not for sets of real numbers but for sequences of real numbers, using an arithmetical denition that involves arithmetical quantification only, and thus is predicative.

In Weyl's system, "analysis" is the predicative version, where the reals are Dedekind cuts, and thus are constructed starting from the naturals.

In contrast, what Weyl calls "hyperanalysis" is a different system where the reals are a new type of objects, in addition to the naturals.

See also Hermann Weyl's Das Kontinuum.

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  • $\begingroup$ I understand that "hyperanalysis" is a system different from the one he spends most of the book outlining. But it is not clear exactly what the difference is. The exegetical challenge here, as I see it, is the last, italized sentence: what hypothesis about what he means by "hyperanalysis" makes that sentence come out true. $\endgroup$
    – Casper
    Jan 10, 2018 at 16:26
  • $\begingroup$ Btw, I think that the identification of reals with Dedekind cuts is orthogonal to the issue here. He does identify his predicative reals with Dedekind cuts, but no matter how far one goes beyond the predicative, one can maintain that identification. $\endgroup$
    – Casper
    Jan 10, 2018 at 16:29

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