.Open SuperStrings Superstrings are a genralization of normal strings - all the same axioms but on a larger class of objects. A superstring (unlike a normal string of symbols like "abc") can hhave `holes` in it -- some subscripts are missing: (1+>"a" | 3+>"c") is a super string. For Sets A, finite A, super(A) ::= Nat<>->A For A, elements(A) ::=Nat>super(A). |()- super(A) generated_by elements(A). ()|-Unique factorisation! For all x:super(A), x={} or for one e:element(A), y:super(A), x=e!y. So define ... first(x)....rest(x)...last(x)..., As always we have |- For P,Q:@super(X), x!P, P!x, P!Q are defined and in @super(A). We have a typical induction schema: SUPER_INDUCTION::=Net{ P::@super(A). {} in P. For all e:elements(A), a!P ==>P ()|- P=super(A). }=::SUPER_INDUCTION. |- SUPER_INDUCTION. We can reduce a superstring to just the elements in a subset of the natural numbers: For N:@Nat, x:super(A), N!x::=N;x. ()|-N!x={ i+>a || i:N and (i+>a) in x }. .Close SuperStrings .Open HyperStrings Replace Nat by Positive Real, hyper(A) ::= { x: Real<>->A || one lub(x) and one glb(x) }, end(x) ::=lub(pre(x)), ... .Close HyperStrings .Open histories histories(A) ::= { x: Real<>->bag(A) || one lub(x) and one glb(x) }, end(x) ::=lub(pre(x)), ... For x,y:histories(A), x!y::@(Real,bag(A))= { (t,e) || t in pre(x)|pre(y) and e=x(t)+y(t) }. .Close histories .Open Super-general A is any set. T is any set S is a subset of @T such that each set t in S has a special maximum value + is an associative operation that preserves maximum values the sum of two T's is greater than the max of them S conatins all the finite subsets of T. supergenstring=S<>->A. .Close Super-general