A survey of binary systems by Bruck R.H.

By Bruck R.H.

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23. This is not itself a Post system, but can be simulated by a Post system. The trick is to add the extra symbol − to the alphabet and combine the rules of the two systems into one large system. In this case the Post-style rules are x x=y , , , |−|=|| |x| x|=|y and |∗|=| , x ∗ y = z, y − z = w x ∗ y = z, x − z = w , x|∗y=w x∗y|=w where x, y, z, w are variables. (Note the use of Post rules with empty premises on the top to get us started. 3 Post systems and computability* 37 Post systems, like the vast majority of other formal systems on finite alphabets, are partially computable.

38 Any infinite finitely branching tree has an infinite path. Proof (Sketch) We consider the set X of all infinite subtrees S of T . This is non-empty as it contains T itself. For the ordering we take, rather unusually, the reverse of ⊆, that is we define S1 S2 if and only if S1 ⊇ S2 . e. a ⊆-minimal subtree) is actually a path. This is like the argument in the previous chapter. If it is not in fact a path and has some branching, then we can find an infinite subtree and hence show the tree is not maximal.

Clearly y1 = y2 but for this y1 we have y1 = u({z ∈ C1 : z < y1 }) = u({z ∈ C2 : z < y2 }) = y2 which is impossible. So this argument shows that there is in fact no element y ∈ C2 \ C1 , and hence that if there is x ∈ C1 which is not in C2 then C2 is an initial segment of C1 . If there is x ∈ C2 which is not in C1 then a similar argument shows C1 is an initial segment of C2 , and if neither of these applies then C1 = C2 . These technical properties of D now complete the proof, for the fact that of any two chains in D one is always an initial segment of the other shows that D = {x ∈ X : there exists C ∈ D such that x ∈ C} is actually a chain.

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