By Harvey M. Friedman

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L2(x) is the unique y such that 2y ≤ x < 2y+1 if x > 0: 0 otherwise. λ2(x) = 2l_2(x) if x > 0; 0 otherwise. 6, i-iv proves every sentence true in M that has only +. , i-iv contains Presburger Arithmetic. Hence π is provably well defined in i-iv, except possibly for l2(x) and λ2(x). Let E = {x: (∀y < x)(2y+1 ≤ 2x)}. Then 0 ∈ E. Let x ∈ E. Since (∀y ≤ x)(2y+1 ≤ 2x+1), we have x+1 ∈ E. We conclude 37 that E is everything. From this, we see that there is at most one y such that 2y ≤ x < 2y+1. Let E = {x: (∃y)(2y ≤ x < 2y+1)} ∪ {0}.

Problem of the straight line as the shortest distance between two points 53 We are not aware of appropriately definite mathematical problems to gauge levels of concreteness. H5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group The modern formulation of this problem is: Are continuous groups automatically differentiable groups? A topological group (continuous group) G is a topological space and group such that the group operations of product and inverse are continuous.

Hence (X,+) is not finitely axiomatizable. 9. (N,<,+) is axiomatized with a single scheme by i. (x+y)+z = x+(y+z), x+y = x+z → y = z. ii. There are unique 0 ≠ 1 such that x+y = 0 ↔ x,y = 0, and x+y = 1 ↔ {x,y} = {0,1}. iii. x < y ↔ x ≠ y ∧ (∃z)(x+z = y). iv. Every definable set containing 0 and closed under +1 is everything. (N,<,+) is not finitely axiomatizable. Proof: Obviously i-iv hold in (N,<,+). Let ϕ hold in (N,<,+). Replace all occurrences of s < t in ϕ by the definition according to iii).

### Boolean Relation Theory and Incompletness (July, 2010 version) by Harvey M. Friedman

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