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Summary A sentence C is independent of a theory T iff neither C, nor the negation of C is derivable from T. A theory is negation-complete iff no sentence in its language is independent of it. Some of key results in metamathematics are independence theorems. According to arithmetical incompleteness theorem, no consistent (recursively axiomatizable) extension of a relatively weak arithmetic is negation-complete. Another important independence result is the independence of the Conituum Hypothesis of the axioms of standard set theory. (There are numerous other examples in analysis, combinatorics, group theory and set theory.) Independence results seem to have impact on philosophical views on mathematical truth and mathematical knowledge. Are sentences independent of mainstream theories determinately true or false and why? If yes, how can we know, which is it? If no, what philosophical views about mathematics are consistent with this view and how are they motivated?
Key works Gödel 1931, Gödel 1940,  Gödel 1947, .Cohen 1963, Feferman manuscript and Feferman et al 2000. For an in-depth study of arithmetical incompletness, see Franzen 2003.
Introductions A great introduction to arithmetical incompleteness theorems and related issues is Smith 2013. A more advanced book is Lindstrom 2002. Franzén 2005 is invaluable. See also Feferman manuscript and Feferman manuscript. As for set-theoretic indeterminacy, see Koellner 2010 and references therein.
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  1. Tim Button (2011). The Metamathematics of Putnam's Model-Theoretic Arguments. Erkenntnis 74 (3):321-349.
    Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.
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  2. Justin Clarke-Doane (2013). What is Absolute Undecidability?†. Noûs 47 (3):467-481.
    It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
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  3. Justin Clarke-Doane, Flawless Disagreement in Mathematics.
    A disagrees with B with respect to a proposition, p, flawlessly just in case A believes p and B believes not-p, or vice versa, though neither A nor B is guilty of a cognitive shortcoming – i.e. roughly, neither A nor B is being irrational, lacking evidence relevant to p, conceptually incompetent, insufficiently imaginative, etc.
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  4. Paul Cohen (1964). The Independence of the Continuum Hypothesis II. Proc. Nat. Acad. Sci. USA 51 (1):105-110.
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  5. Paul Cohen (1963). The Independence of the Continuum Hypothesis. Proc. Nat. Acad. Sci. USA 50 (6):1143-1148.
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  6. Solomon Feferman, Is the Continuum Hypothesis a Definite Mathematical Problem?
    The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.
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  7. Harvey M. Friedman, Concrete Mathematical Incompleteness.
    there are mathematical statements that cannot be proved or refuted using the usual axioms and rules of inference of mathematics.
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  8. Han Geurdes, The Construction of Transfinite Equivalence Algorithms.
    Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical (...)
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  9. Adrian Riskin (1994). On the Most Open Question in the History of Mathematics: A Discussion of Maddy. Philosophia Mathematica 2 (2):109-121.
    In this paper, I argue against Penelope Maddy's set-theoretic realism by arguing (1) that it is perfectly consistent with mathematical Platonism to deny that there is a fact of the matter concerning statements which are independent of the axioms of set theory, and that (2) denying this accords further that many contemporary Platonists assert that there is a fact of the matter because they are closet foundationalists, and that their brand of foundationalism is in radical conflict with actual mathematical practice.
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  10. Raymond Smullyan (1996). Set Theory and the Continuum Problem. Clarendon Press.
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