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Topic: Axiomization of Number Theory
Replies: 52   Last Post: Aug 4, 2003 12:18 AM

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 Mike Oliver Posts: 1,518 Registered: 12/6/04
Re: Axiomization of Number Theory
Posted: Jul 30, 2003 11:53 PM

Andrew Boucher wrote:
> David C. Ullrich wrote:
>> Supposing for the sake of argument that Keith was correct
>> when he said that there's no formal system for second-order
>> logic corresponding to the well-known formal systems for
>> first-order logic, exactly what does it mean for something
>> to go through in second-order PA?

>
> It means, can it be proved in the system PA2, second-order Peano
> Arithmetic? Briefly, PA2 is a system of second-order logic with full
> comprehension and the Peano Axioms (where the Induction Schema may be
> replaced by an axiom).

This is a spot where it's easy to get confused over terminology, and
I'm not even sure exactly what the standard terminology is. But
in any case what Andrew is describing (which I snipped) is actually
a theory of two-sorted *first*-order logic. Its models have (in one
way of describing them) two universes: One over which the
natural-number variables range, and another over which the set-of-naturals
variables range. I think these may be called "Henkin models" or
some such.

PA2, interpreted *this* way, *does* admit a formal system that mechanically
verifies or rejects putative proofs, and it does *not* fully determine
the natural numbers up to isomorphism (or even decide all first-order

When you're talking about true second-order logic, you can take exactly
the same axioms as above, and now they *do* determine the natural numbers
up to isomorphism. But a "model" is a different sort of gadget: It
has only one universe. The natural-number variables range over elements
of that universe, and the set variables range over subsets of the universe.

Date Subject Author
7/25/03 Charlie-Boo
7/25/03 Charlie Johnson
7/25/03 Arief
7/25/03 Jeffrey Ketland
7/28/03 Charlie-Boo
7/28/03 William Elliot
7/28/03 Charlie-Boo
7/28/03 Andrew Boucher
7/29/03 Andrew Boucher
7/29/03 Andrew Boucher
8/4/03 Charlie-Boo
7/29/03 Pete Moore
7/29/03 Robin Chapman
7/29/03 David C. Ullrich
7/29/03 Robin Chapman
7/29/03 David C. Ullrich
7/29/03 Robin Chapman
7/30/03 David C. Ullrich
7/30/03 Robin Chapman
7/30/03 David C. Ullrich
7/31/03 Robin Chapman
7/29/03 George Cox
7/29/03 Per Eriksson
7/30/03 Charlie-Boo
7/30/03 Per Eriksson
7/31/03 Robin Chapman
7/30/03 David C. Ullrich
7/30/03 Per Eriksson
7/30/03 Mike Oliver
7/31/03 David C. Ullrich
7/31/03 Charlie-Boo
7/31/03 tchow@lsa.umich.edu
7/31/03 Per Eriksson
7/29/03 Arief
7/30/03 Keith Ramsay
7/30/03 David C. Ullrich
7/30/03 Andrew Boucher
7/30/03 David C. Ullrich
7/30/03 Andrew Boucher
7/30/03 Mike Oliver
7/31/03 Andrew Boucher
7/31/03 Mike Oliver
7/31/03 Aatu Koskensilta
7/31/03 Andrew Boucher
7/31/03 Mike Oliver
7/31/03 Andrew Boucher
7/31/03 Mike Oliver
7/31/03 Andrew Boucher
7/31/03 Mike Oliver
7/31/03 Aatu Koskensilta
7/31/03 Andrew Boucher
7/31/03 Keith Ramsay
7/31/03 Andrew Boucher