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> Accounts of Pressburger Arithmetic, and the proof that arithmetic with
> addition (and successor) but NOT multiplication are decidable, are often
> preceded by a comment or so about arithmetic with "only one of"
multiplication
> or addition.

> Now I know of the standard way that addition can be *defined* in terms of
> multiplication and successor, but I’m not sure that this has too much to
do
> with the oft-suggested implication that "arithmetic with multiplication
only
> is decidable".

> Is that last statement ever made, explicitly?  I don’t recall it being so.

> And just what would it BE – arithmetic with multiplication and successor,
> but not addition.  What sensible axioms could it be given?

> Help please.

A good reference for decidability of Th(N;=,x) (the elementary theory of
integer multiplication – sometimes called "Skolem Arithmetic") is :
C.Smorynski, "Logical Number Theory I", Springer-Verlag,  1991.
It presents Cegielski’s proof (1981) by quantifier elimination.
Alternative proofs were given by Mostowski ("On directs products of
theories",J.Symb.Log. 17, 1-31, 1952) and later Hodgson ("On direct products
of automaton decidable theories", Theor.Comp.Sci. 19,331-335, 1982) — both
use Presburger’s decidability result for Th(N;=,+).
Alexis Bes

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Bill Taylor wrote:
> Accounts of Pressburger Arithmetic, and the proof that arithmetic with
> addition (and successor) but NOT multiplication are decidable, are often
> preceded by a comment or so about arithmetic with "only one of" multiplication
> or addition.

> Now I know of the standard way that addition can be *defined* in terms of
> multiplication and successor, but I’m not sure that this has too much to do
> with the oft-suggested implication that "arithmetic with multiplication only
> is decidable".

> Is that last statement ever made, explicitly?  I don’t recall it being so.

> And just what would it BE – arithmetic with multiplication and successor,
> but not addition.  What sensible axioms could it be given?

> Help please.

> ——————————————————————————-
>              Bill Taylor            W.Tay…@math.canterbury.ac.nz
> ——————————————————————————-
> Godel’s humility theorem:-
>                             Providing you are capable of doing sufficient math,
>                    you are intelligent if and only if you cannot prove you are.
> ——————————————————————————-

    Goedel’s complementary humility theorem:
                       Providing you are capable doing sufficient physics,
                       you are intelligent if and only if you can prove you
                       are ignorant of Goedel’s theorem.

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Bill Taylor wrote:

> Accounts of Pressburger Arithmetic, and the proof that arithmetic with
> addition (and successor) but NOT multiplication are decidable, are often
> preceded by a comment or so about arithmetic with "only one of" multiplication
> or addition.

> Now I know of the standard way that addition can be *defined* in terms of
> multiplication and successor, but I’m not sure that this has too much to do
> with the oft-suggested implication that "arithmetic with multiplication only
> is decidable".

> Is that last statement ever made, explicitly?  I don’t recall it being so.

> And just what would it BE – arithmetic with multiplication and successor,
> but not addition.  What sensible axioms could it be given?

Consider–add to suitable* axioms for first order logic these:
1. if (x = y) then (if (x = z) then (y = z))
2. if (x = y) then (sx = sy)
3. 0 <> sx
4. if (sx = sy) then x = y
5. x + 0 = x
6. x + sy = s(x + y)
9A. for any wff A, if A0 then (if (all x)(if Ax then Asx) then (all
x)Ax)
Then you have Presburger’s 1929 axioms.
Now replace those with:
1. if (x = y) then (if (x = z) then (y = z)
2. if (x = y) then (sx = sy)
3. 0 <> sx
4. if (sx = sy) then x = y
7. x.0 = 0
8n. for each natural number n, N0.(sy) = N(N0.y), where "N" is n ‘s’s.
9A. for any wff A, if A0 then (if (all x)(if Ax then Asx) then (all
x)Ax)
and you have _my_ candidate for arithmetic without +.
8n is just an axiom schema to replace
8. x.(sy) = (x.y) + x
by replacing "+ x" with x-many successor function applications.
Could be rubbish–no guarantee of fitness for purpose!
*Note, 1. is needed because the "suitable" fol has no axioms for "=", it
is introduced as a binary predicate.
All of these "s"s remind me to remind you that "Presburger" only has
one.

Regards, PP

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> Help please.

> ——————————————————————————-
>              Bill Taylor            W.Tay…@math.canterbury.ac.nz
> ——————————————————————————-
> Godel’s humility theorem:-
>                             Providing you are capable of doing sufficient math,
>                    you are intelligent if and only if you cannot prove you are.
> ——————————————————————————-

Now I know of the standard way that addition can be *defined* in terms of
multiplication and successor, but I’m not sure that this has too much to do
with the oft-suggested implication that "arithmetic with multiplication only
is decidable".

Is that last statement ever made, explicitly?  I don’t recall it being so.

And just what would it BE – arithmetic with multiplication and successor,
but not addition.  What sensible axioms could it be given?

Help please.

——————————————————————————-
             Bill Taylor            W.Tay…@math.canterbury.ac.nz
——————————————————————————-
Godel’s humility theorem:-
                            Providing you are capable of doing sufficient math,
                   you are intelligent if and only if you cannot prove you are.
——————————————————————————-