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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.

P.S.

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

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)