Is there an algorithm for deciding which statements of number theory are true? - Leibniz’s Question

Can the consistency of number theory be proved using only non-dubious principles of finitary reasoning? - Hilbert’s Question

Section 1 defines the URM.

The URM can store an infinite number of positive integers in registers R1 upto Rn and has the following instruction set:

( Name: Notation ; Effect )

Zero: Z(n) ; Replace the number in R(n) by 0.

Successor: S(n) ; Add 1 to the number in R(n).

Copy : C(m,n) ; Replace number in R(n) by number in R(m).

Jump : J(m,n,q) ; If R(m) = R(n) jump to q otherwise next.

From this limited instruction set a powerful programming language can be constructed.

Section 2 investigates which functions can be implemented on a URM.

The following URM program for example

1 J(1,3,12)

2 J(2,3,11)

3 C(1,3)

4 S(5)

5 J(2,5,11)

6 Z(4)

7 S(3)

8 S(4)

9 J(1,4,4)

10 J(1,1,7)

11 C(3,1)

implements -multiplication- on a URM.

So far. To be continued next week.

P.S.

Trivia:

- Russell's Paradox. A is the set of sets which do not contain A.

- Russell and Whitehead proved that 1+1=2: more on this page.

## No comments:

## Post a Comment