About this topic
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 2012. 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.
Related categories

15 found
  1. The Metamathematics of Putnam's Model-Theoretic Arguments.Tim Button - 2011 - 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.
  2. Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
  3. What is Absolute Undecidability?†.Justin Clarke-Doane - 2013 - 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 (...)
  4. Flawless Disagreement in Mathematics.Justin Clarke-Doane - unknown
    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.
  5. The Independence of the Continuum Hypothesis II.Paul Cohen - 1964 - Proc. Nat. Acad. Sci. USA 51 (1):105-110.
  6. The Independence of the Continuum Hypothesis.Paul Cohen - 1963 - Proc. Nat. Acad. Sci. USA 50 (6):1143-1148.
  7. Models and Computability.W. Dean - 2014 - Philosophia Mathematica 22 (2):143-166.
    Computationalism holds that our grasp of notions like ‘computable function’ can be used to account for our putative ability to refer to the standard model of arithmetic. Tennenbaum's Theorem has been repeatedly invoked in service of this claim. I will argue that not only do the relevant class of arguments fail, but that the result itself is most naturally understood as having the opposite of a reference-fixing effect — i.e., rather than securing the determinacy of number-theoretic reference, Tennenbaum's Theorem points (...)
  8. Math Anxiety.Aden Evens - 2000 - Angelaki 5 (3):105 – 115.
    This article presents an explication of the references to the history of the calculus in the first few pages of Chapter 4 of Deleuze's _Difference and Repetition_. In those pages, Deleuze uses anachronistic readings of the calculus to explain his theory of ontogenesis, beginning with the differential, dx, that is strictly nothing by itself but that establishes singular points in relation to other differentials. He builds from the differential to power series, showing a corresponding process of determination in the ontogenesis (...)
  9. Is the Continuum Hypothesis a Definite Mathematical Problem?Solomon Feferman - manuscript
    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.
  10. Concrete Mathematical Incompleteness.Harvey M. Friedman - unknown
    there are mathematical statements that cannot be proved or refuted using the usual axioms and rules of inference of mathematics.
  11. The Construction of Transfinite Equivalence Algorithms.Han Geurdes - manuscript
    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 (...)
  12. Can the Cumulative Hierarchy Be Categorically Characterized?Luca Incurvati - 2016 - Logique Et Analyse 59 (236):367-387.
    Mathematical realists have long invoked the categoricity of axiomatizations of arithmetic and analysis to explain how we manage to fix the intended meaning of their respective vocabulary. Can this strategy be extended to set theory? Although traditional wisdom recommends a negative answer to this question, Vann McGee (1997) has offered a proof that purports to show otherwise. I argue that one of the two key assumptions on which the proof rests deprives McGee's result of the significance he and the realist (...)
  13. On the Most Open Question in the History of Mathematics: A Discussion of Maddy.Adrian Riskin - 1994 - 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.
  14. Set Theory and the Continuum Problem.Raymond Smullyan - 1996 - Clarendon Press.
  15. A Metasemantic Challenge for Mathematical Determinacy.Jared Warren & Daniel Waxman - forthcoming - Synthese:1-19.
    This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly popular approach (...)