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THE MATHEMATICAL MEANING OF MATHEMATICAL LOGIC
by
Harvey M. Friedman
[email protected]
http://www.math.ohiostate.edu/~friedman/
April 15, 2000
Rev. April 21, 2000
I am going to discuss the mathematical meaning of
1. the completeness theorem.
2. the incompleteness theorems.
3. recursively enumerable sets of integers.
4. constructivity.
5. the Ackerman hierarchy.
6. Peano arithmetic.
7. predicativity.
8. Zermelo set theory.
9. ZFC and beyond.
Each of these theorems and concepts arose from very specific
considerations of great general interest in the foundations
of mathematics (f.o.m.). They each serve well defined
purposes in f.o.m. Naturally, the preferred way to formulate
them for mathematical logicians is in terms that are close
to their roots in f.o.m.
However, the core mathematician does not come out of the
f.o.m. tradition as does the mathematical logician. Instead,
he/she comes out of the much older arithmetic/
algebraic/geometric (a.a.g.) tradition. The significance of
these theorems and concepts are not readily apparent from the
a.a.g. point of view.
In fact, a full formulation of these theorems and concepts
requires the introduction of rather elaborate structures
which can only be properly appreciated from a distinctly
f.o.m. perspective. In fact, the a.a.g. perspective is of
little help in gaining facility with these elaborate
structures.
So the core mathematician, steeped in a.a.g, is very unlikely
to spend the considerable effort required to understand the
meaning of such theorems and concepts. In wading through
these developments, he/she will not be putting the a.a.g.
perspective to effective use, and will not anticipate any
corresponding proportionate a.a.g. payoff.
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Of course, there is nothing to prevent a core mathematician
from becoming familiar with and being perfectly comfortable
with the f.o.m. tradition. But for various reasons, this has
become quite rare.
To give a prime example of what I have in mind, most of these
theorems and concepts depend on the syntax and semantics of
so called first order predicate calculus with equality. This
is a rather elaborate structure which, with proper
substantial and detailed discussion, sounds like beautiful
music to the ears of an f.o.m. oriented listener  but more
like painful, long winded noise to many others.
So in this talk, I want to give
relatively
a.a.g. friendly
presentations of these fundamental theorems and concepts from
f.o.m. I say
relatively
because I do not attempt to go all
the way here. One can go much further. But I do go far enough
in the direction of a.a.g. friendliness that the mathematical
meaning of these presentations should be apparent to this
audience. Bear in mind that the project of systematically
giving such a.a.g. friendly treatments is, as far I know,
quite new, and raises substantial issues  both technical and
conceptual  about which I know very little at the present
time.
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 Fall '08
 JOSHUA
 Math, Logic, Mathematical logic, free variable statement

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