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Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
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Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.
Polynomial equations with quantifierprefixes
, 2010
"... A prefixed polynomial equation (or “a polynomial expression with a quantifierprefix”) 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 ..."
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A prefixed polynomial equation (or “a polynomial expression with a quantifierprefix”) 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: 1Con(IΣ1)); a polynomial expression equivalent to PH 3 (strength: 1Con(IΣ2)); a polynomial expression equivalent to PH (strength: 1Con(PA)); a polynomial expression equivalent to KT (strength: 1Con(ATR0)); a polynomial expression for KTr·log (a phase transition between EFAprovability and ATR0unprovability); a polynomial expression for the Graph Minor Theorem (strength: at least 1Con(Π 1 1CA0)); a polynomial expression for planar GMT (a phase transition between EFAprovability 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 BAFterms; a polynomial expression equivalent to Proposition E of Boolean Relation Theory (strength: 1Con(ZFC + {nMahlo}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 1Con(ZFC + {nMahlo 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 setup the answers become quite short: often 14 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