[FOM] PA inconsistencies

[Thomas Klimpel]

> >> FOM Readership: what are the main references for "mathematics without
> >> exponentiation" and how far has it gotten?
> >
> > there is a large literature on bounded arithmetic and related questions.
>
> I meant an depth development of actual mathematics, (analysis,
> algebra, number theory, differential equations, topology, geometry,
> group theory, etc,.), not computational complexity developments or
> math logic developments.

Why exclude math logic developments from this question? They might be
simple to state and understand and yet extremely surprising. At least
I was totally surprised when I learned that Robinson’s arithmetic +
induction for bounded existential formulas has no computable
nonstandard models
(https://mathoverflow.net/questions/38160/computable-nonstandard-models-for-weak-systems-of-arithemtic).
In fact, even the additive reduct (M,+,<=) is recursively saturated
(https://mathoverflow.net/a/295115).

Excluding computational complexity seems even stranger to me. The
point of doing actual mathematics in those systems would be to ensure
that the required computations are practically feasible, and this is
also part of what computational complexity tries to achieve. In
addition, it tells you why the simple encoding techniques usable in
EFA run into trouble, and how to fix that: "V^0 is capable of talking
about sequences using a better encoding than Godel’s beta function
(write the numbers in the sequence in binary, add 2 between each
consecutive pair, read in base 4)."
(https://golem.ph.utexas.edu/category/2011/10/weak_systems_of_arithmetic.html#c039690)

There are people like NJ Wildberger, which have been previously
discussed on this list
(https://cs.nyu.edu/pipermail/fom/2012-July/016563.html), that take
their own take on ultrafinitism seriously, and still manage to teach
nearly normal introductory (and advanced) courses on a variety of
subjects (Linear Algebra, Probability and Statistics, Elementary
Mathematics, Differential Geometry, Algebraic Topology, History of
Mathematics, Famous Math Problems) in addition to lecturing on their
hobby horses (Foundations of Mathematics, Rational Trigonometry,
Universal Hyperbolic Geometry). You might object that those are just
videos (https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ), but
NJ Wildberger also produces normal articles and text books
(http://web.maths.unsw.edu.au/~norman/). He proposed his own new
logical principle: "Don’t pretend that you can do something that you
can’t." I tried to discuss with him about limitations of that point of
view (https://njwildberger.com/2015/11/27/a-new-logical-principle/#comment-1057),
but apparently he was not interested in that sort of discussion.