[FOM] Alain Connes' approach to Analysis

[Joe Shipman]

I don’t understand why suspicion of nonmeasurable sets and ultrafilters should make a number theorist distrust proofs which use them, it is well known that they can be constructively eliminated from the proof of any Arithmetical statement.

Is “there exists a non-principal ultrafilter on P(N)” strictly weaker than “every filter on P(N) can be extended to an ultrafilter”? Is the latter statement strictly weaker than “there exists a well-ordering of the continuum”?

— JS

Sent from my iPhone

> On Sep 16, 2018, at 8:16 AM, Mikhail Katz <katzmik at macs.biu.ac.il> wrote:
> 
> I hesitated answering this particular post on Connes' criticism of
> Robinson and the hyperreals, since I was hoping somebody from the
> reverse math community would respond.  Since the thread seems to be
> still officially open, perhaps a reply is in order.  I would like to
> mention the following points.
> 
> (1) Connes' critique of the hyperreals on the grounds that they "lead
> canonically to a nonmesurable set" seem to have some kind of a
> prurient appeal and refuses to fade away and therefore some
> clarifications seem appropriate.
> 
> (2) The fact is that a nonstandard integer H in *N leads "canonically"
> to an ultrafilter {A: H\in *A} on N.
> 
> (3) An ultrafilter naturally leads to a nonmesurable set.
> 
> (4) Therefore the criticism of "canonically leading to nonmeasurable
> sets" actually targets Tarski (rather than Robinson), who invented
> ultrafilters in 1930 (or Bourbaki, who invented ultrafilters in 1935
> as believed by many in France including Connes).
> 
> (5) Connes routinely uses ultrafilters in many of his own articles and
> books, without mentioning anything about their leading to
> nonmeasurable sets.
> 
> (6) This occurs also in Connes' papers and books that voice the
> "nonmeasurable set" criticism against Robinson.
> 
> (7) Skolem's nonstandard integers embed in *N.  Hence by Connes'
> logic, a Skolem nonstandard integer also leads to a nonmeasurable set.
> 
> (8) Yet Skolem's construction takes place in ZF (without choice).
> 
> (9) What this illustrates is the POWER of Robinson's transfer
> principle that stands behind item (1) above, rather than any WEAKNESS
> of his framework.
> 
> (10) Connes' claim to the contrary amounts to an attempt to dress down
> a feature to look like a bug, to reverse a familiar quip from software
> developers.
> 
> 
> On 3 september 2018, Jose Caballero wrote:
> 
> "Connes criticizes Leibnitz's approach as follows (free translation
> from French): Every nonstandard real number determines, in a canonical
> way, a subset of [0,1] which is not Lebesgue measurable, hence such
> numbers do not exist. Reference (page 6) in Kanovei-Katz-Mormann's
> paper: https://arxiv.org/pdf/1211.0244.pdf"
> 
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