[FOM] R: stopping at ACA_0
drago at unina.it
Sun Feb 19 17:42:37 EST 2006
> Harvey Friedman wrote:
> > Any story you can make about some precise stopping place for
> > predicativity, and I can make a much better story about stopping
> > at ACA_0.
> Nik Weaver replied:
> I like ACA_0 and am genuinely interested
> in knowing whether it really has a "nice story" so I'll ask again.
In a previous posting I recalled that Feferman's studies located the very
important Weyl's elementary mathematics at the level of ACA_0. The
characteristic features of Weyl mathematics are equivalently:
1) geometrical intuition of single point of intersection, of parallelism,
2) the use of one quantifier only upon rational numbers,
3) there exist the l.u.b (or the g.l.b.) when there exists an approximating
sequence to it.
I suggested that the same characteristic features hold true in Cavalieri's
theory of indivisibles, which first introduced:
the notion of infinity in science
calculus (in Weyl's mathematics)
and a quantifier (Cavalieri's latin word Omnes, representign te integration,
is exactly translated in English by the word all).
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Da: "" <nweaver at math.wustl.edu>
A: <fom at cs.nyu.edu>
Data invio: venerdì 17 febbraio 2006 7.07
Oggetto: [FOM] stopping at ACA_0
> > On can stop a lot earlier than "predicativity", say, stop at
> > ACA0 or RCA0, or one can stop somewhat higher than "predicativity",
> > say at one inductive definition, or Pi11-CA0. Or one can stop even
> > higher at, say, the theory of a recursively inaccessible, or what
> > have you.
> > All of these stopping places, and many more, have very "nice"
> > stories. All of these stories have advantages and disadvantages.
> > These advantages and disadvantages make sense and have their
> > proponents, both mathematically and philosophically.
> In response I asked
> > I have to admit I've never heard of anyone advocating ACA_0 as
> > a basic philosophical stance. Yet you tell me there is a "nice
> > story" in its favor, which "has proponents both mathematically
> > and philosophically". Can you say who some of those proponents
> > are, and what that nice story is, in the case of ACA_0?
> and got the non-answers
> > Child's play to come up with one that looks as attractive as
> > tortuous involved controversial ones for predicativity.
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