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**1 - 2**of**2**### Polynomial equations with quantifier-prefixes

, 2010

"... A prefixed polynomial equation (or “a polynomial expression with a quantifier-prefix”) is an equation P (x1, x2,..., xn) = 0, where P is a polynomial whose variables x1, x2,... xn range over natural numbers, that is preceded by some quantifiers over some or all of its variables x1, x2,..., xn. Poly ..."

Abstract
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A prefixed polynomial equation (or “a polynomial expression with a quantifier-prefix”) is an equation P (x1, x2,..., xn) = 0, where P is a polynomial whose variables x1, x2,... xn range over natural numbers, that is preceded by some quantifiers over some or all of its variables x1, x2,..., xn. Polynomial equations P (x1, x2,..., xn) = 0 without quantifiers are also called Diophantine. We study prefixed polynomial equations and produce a range of unprovable statements of this form:- a polynomial expression equivalent to PH 2 (strength: 1-Con(IΣ1));- a polynomial expression equivalent to PH 3 (strength: 1-Con(IΣ2));- a polynomial expression equivalent to PH (strength: 1-Con(PA));- a polynomial expression equivalent to KT (strength: 1-Con(ATR0));- a polynomial expression for KTr·log (a phase transition between EFA-provability and ATR0-unprovability);- a polynomial expression for the Graph Minor Theorem (strength: at least 1-Con(Π 1 1-CA0));- a polynomial expression for planar GMT (a phase transition between EFA-provability and the full strength of GMT);- a polynomial expression that knows all values of all polynomials;- a polynomial expression that knows all values of all BAF-terms;- a polynomial expression equivalent to Proposition E of Boolean Relation Theory (strength: 1-Con(ZFC + {n-Mahlo}n∈ω). This early draft only has a long version (which will shrink in later drafts!) In this early draft, the lengths of polynomials haven’t yet been reasonably minimised. So far we only aimed for the very first very coarse polynomials. The results below will be improved before the end of summer 2010: all polynomials will be shortened, and the polynomial of Chapter 7 for 1-Con(ZFC + {n-Mahlo cardinals}n∈ω) will be shortened very considerably. A second (not yet typed) layer of this project is to write expressions that allow exponentiation x y and logarithm to be mentioned. In this more expressive set-up the answers become quite short: often 1-4 lines. We estimate that even Proposition E will fit well onto 13 lines or less. The currently growing part of the draft is the theory of seeds (polynomial equations of minimal length in their