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only if every nondecreasing computable function fromthe unit interval to the real numbers is differentiable at r. ... Now we can formulate the announced result. Theorem. A real number r is normal to a given base b if, and only if, every real valued

The strictest version, FAN or Decidable Fan, is tosay that the bar B in question is decidable: every node is either in B or not.Another natural version, FANΠ01 or Π. ... M|= FAN. Proof. If B is a decidable bar in the ground (realizability) model, then M

Suppose γ Sand take B(β,γ) satisfying (i) and (ii). Then Bα Aγ is finite for every α < γ, α S andtherefore. ... Then we choose a decreasing sequence (ank)k such that(a) ank an0 , ank yn for every k(b) limk λ(a.

dk(W) =. W ′DkBn,W ′W. diam(W ′)s. diam(W)s= 1. Claim 3.5 For every k N, U B, with diam(U) > 0, dk(U) c1s2k. ... 1s2kdiam(W)s. Since for every U B there is a W B0 with U W we have that.

For the p-computation of d′, let b be such that for every r,. ... yn| 12d(xn). 7. Therefore if α reall(S[d])) then Sl(α) S[d] and since for every n,.

i) There exist a,b > 0 such that a ‖en‖,‖en‖ b for every n N,(ii) ek(en) = δkn for every k,n N,. ... We say that B is monotone whether Kb = 1. Along the paper we will refer to every such B simply as a basis.

Acknowledgments 9References 9. 2. Basics on lattice theory. A lattice (L,,) is a poset L in which every two elements a,b havea unique supremum (or join) a b and a ... every element can be written uniquely as a join of atoms.It follows that if [a,b] and [c

ω} such that, for every B, if {dege(ΘBr ) : r ω} is not computably inde-pendent, then B 02. ... Claim 5. For every B and σ B, j(σ) ΦΘB6=σ. σ if and only if j(σ) gets eventuallyrealized.

Actually, they prove a bound on the diameter of any poly-tope { x IRn | A x b, b IRm } where A is a totally unimodular matrix. ... jk, ik, b) of length two less than P. Repeatingthis procedure will decrease the length of the path between a and b by two

Since C is locally compact, every definable set is Fσ. Thesame is true for the complement, so every definable set is also Gδ. ... If (K, , ,E) |= Φ, then every definable subsetof K is countable or co-countable.