In an interview (at <http://www.alainconnes.org/docs/Inteng.pdf>) Connes remarks that



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> I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point is that as soon as you have a non-standard number, you get a non-measurable set. And in Choquet’s circle, having well studied the Polish school, we knew that every set you can name is measurable; so it seemed utterly doomed to failure to try to use non-standard analysis to do physics. 
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What does he mean; what is he referring to?



A remark of Connes on non-standard analysis

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> ...as soon as you have a non-standard number, you get a non-measurable set.
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Every nonstandard natural number $N$ gives rise to a nonprincipal ultrafilter $U$ on $\mathbb{N}$, by saying that a set $X\subset\mathbb{N}$ is in $U$ if and only if $N\in X^\*$, the nonstandard analogue of $X$. In other words, the ultrafilter says $X$ is large if it expresses a property that the nonstandard number $N$ has. We may regard $U$ as a subset of $2^\mathbb{N}$, which carries a natural probability measure. But a nonprincipal ultrafilter cannot be measurable there, since the full bit-flipping operation, which is measure-preserving, carries $U$ exactly to its complement, so $U$ would have to have measure $\frac12$, but $U$ is invariant by the operation of flipping any finite number of bits, and so must have measure $0$ or $1$ by [Kolmogorov's zero-one law](https://en.wikipedia.org/wiki/Kolmogorov%27s_zero-one_law). (See also [this article by Blackwell and Diaconis](https://projecteuclid.org/ebooks/institute-of-mathematical-statistics-lecture-notes-monograph-series/Statistics-probability-and-game-theory/chapter/A-non-measurable-tail-set/10.1214/lnms/1215453560) proving the same fact.)



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Another way of saying that a set is easily described is to say that it lies low in the [descriptive set-theoretic hierarchy](https://en.wikipedia.org/wiki/Descriptive_set_theory), and the lowest such sets are necessarily measurable. For example, every set in the Borel hierarchy is measurable, and the Borel context is often described as the domain of *explicit* mathematics. Under stronger set-theoretic axioms, such as large cardinals or PD, the phenomenon rises to higher levels of complexity, for under these hypotheses it follows even that all sets in the projective hierarchy are Lebesgue measurable. This would include any set that you can define by quantifying over the reals and the integers and using any of the basic mathematical operations.


