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Saturday, January 15, 2011

[M381-Logic] - #2: Unit 1

Already comfortable with the notion that Saturday is "M381-Logic-day". The purpose of the Logic half of M381 is finding answers to the following two questions.

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.


- 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.

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