By J. Barwise

Booklet through Barwise, J.

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**Example text**

If r ~ a x a and r is well founded on a then we say that r is well founded. \ (y,X)Er}, for all XEa. The function f is said to be collapsing for r. \ f is collapsing for r]. ) Beta is not provable in KPU but it IS provable if we add :EI Separation. 6 Theorem. Beta is provable in KPU + (:E I Separation). Sketch of proof. Let us work in K PU + (:EI Separation). Let r be well founded on a, and write x-

We check three axioms and trust the student to verify the other two, Extensionality and Union. lJIl=cx=rk(x) 1\ fJ =rk(y). lJl. lJll=b={x,y} byabsoluteness of the formula b = {x, y}, from Table 1. lJl. Let cp(x,y) be L\o. lJI. lJl. lJl by absoluteness. lJl. lJIl=cp(x,y)l\rk(y)=cx. lJIl=rk(y)=cx, Thus in ~iR we have VXEa 3cx 3y[rk(y)=cx 1\ cp(x,y)]. lJI such that VXEa 3cx

9 Definitions. VI=(9Jl;A,E, ... ). We often use the following notation and terminology. An object x is in b.. if xEMuA, and we write XEb... A relation on b.. is a relation on MuA. An n-ary relation S on b.. is ~1 on b.. , such that for all x 1, ... ,X nEb... The relation S is ill on b.. 11 on b.. if S is both ~1 and ill on b... A function F on b.. is a function with domain a subset of (MuA)" for some n and range a subset of MuA. We say F is ~1 on b.. if its graph is ~1 on b... 10 Proposition. Let b..