By Kripke, Saul
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Additional resources for Elementary Recursion Theory and its Applications to Formal Systems
X ≤ y) (x ∈ y ⊃ B); (∃x ∈ y) B =df. (∃x ≤ y) (x ∈ y ∧ B). Note also that if n codes a set S and S' ⊆ S, then there is an m ≤ n which codes S'. ) We can therefore define x ⊆ y =df. (z ∈ x) z ∈ y; (x ⊆ y) B =df. (x ≤ y) (x ⊆ y ⊃ B); (∃x ⊆ y) B =df. (∃x ≤ y) (x ⊆ y ∧ B). Now that we can code finite sets of numbers, it is easy to code finite sequences. e. sets of ordered pairs, which we can in turn identify with sets of numbers, since we can code up ordered pairs as numbers. ) Finally, those sets can themselves be coded up as numbers.
It is usually emphasized as a basic requirement of logic that the set of formulae of a given language must be decidable, but it is not clear what the theoretical importance of such a requirement is. Chomsky’s approach to natural language, for example, does not presuppose such a requirement. In Chomsky's view, a grammar for a language is specified by some set of rules for generating the grammatically correct sentences of a 33 Elementary Recursion Theory. Preliminary Version Copyright © 1996 by Saul Kripke language, rather than by a decision procedure for grammatical correctness.
The third disjunct will be a formula true of two numbers a,s if a is an atomic formula of the form P23t1t2t3, where t1, t2 and t3 are terms of RE, and s is sufficient for and satisfies a: D3(a,s)=df. (∃m1≤s)(∃m2≤s)(∃m3≤s)(Seql(s,0(4)) ∧ Term(m1) ∧ Term(m2) ∧ 40 Elementary Recursion Theory. Preliminary Version Copyright © 1996 by Saul Kripke Term(m3) ∧ [0(1),[0(5),[0(3),0(2)]]]∈s ∧ [0(2),m1]∈s ∧ [0(3),m2]∈s ∧ [0(4),m3]∈s ∧ (∃y1≤a)(∃y2≤a)(∃y3≤a)(Den(m1,y1,s) ∧ Den(m2,y2,s) ∧ Den(m3,y3,s) ∧ Μ(y1,y2,y3)).