By Harvey M. Friedman
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Additional resources for Boolean Relation Theory and Incompletness (July, 2010 version)
Dx ≠ 1. e. d|x ∨ d|x+1 ∨ d|x+1+1 ∨ ... +1 with d disjuncts. where d ≥ 2. ). 1. Accordingly, fix a positive integer t, let K consist of a) and those instances of b-e based on 2 ≤ d ≤ t. Let K' be the result of applying π, and then existentially quantifying out the constant 1. We can pull out the conjuncts emanating from a)-c) since they do not mention 1. We claim that the result of applying π to a)-c), which just means applying π to b), is provable in i-v. This is obvious for a),b). For c), suppose dx = dy ∧ x ≠ y.
The answer is yes by a simple construction. Let F,K be a counterexample. Build an appropriate infinite sequence from F and from K, and use the subfield of F generated by the infinite sequence from F. Nagata gave a negative answer to H14 in [Na59]. But there are other related problems that have either been affirmed or are open. Can we also a priori restrict to countable systems? H15. Rigorous foundation of Schubert's enumerative calculus Hermann Schubert claimed some spectacular counts on the number of geometric objects satisfying certain conditions, using methods that do not meet current rigorous standards even in 1900.
Its intelligibility arguably depends on some understanding of abstract set theory. But nevertheless, with the help of coding, it asserts the nonexistence of a finite configuration. In contrast, AxC and CH cannot be formulated in this way, regardless of coding devices. These statements live inherently in the abstract set theoretic universe. Subsequent developments in Incompleteness initially centered around analyzing a large backlog of problems from abstract set theory, mostly with the help of Cohen's method of forcing introduced in [Co63,64].
Boolean Relation Theory and Incompletness (July, 2010 version) by Harvey M. Friedman