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      Questions tagged [lo.logic]

      first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.

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      Survey article model theory research

      I've taken a graduate course in model theory and I like it so much that I can imagine doing research in this area. Are there survey articles or review papers on the current research topics in model ...
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      1answer
      141 views

      A weak (?) form of Shelah cardinals

      The following definition of a large cardinal property combines parts of the definitions of "Shelah cardinal" and "Woodin cardinal": A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \...
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      Question about relation of admissibility and cofinality (of infinite-time functions)

      Due to its nature, this is a very long question (so please bear with me). To convey the question, while keeping it in manageable length, I have tried to convey the main idea in the question (rather ...
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      1answer
      50 views

      On Asymptotic classes of finite structures (2)

      As I mentioned in my previous question, I am reading the following paper: One-dimensional asymptotic classes of finite structures. by: Macpherson and Steinhorn I have some crazy questions! What is ...
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      1answer
      178 views

      Is the power set axiom essential for constructing L?

      Take ZFC, remove axiom of Power set, and put instead of it the following axiom: Axiom of Successor Cardinals: $\forall \kappa \exists x \forall \alpha ( \alpha \leq \kappa \to \alpha \in x)$ where "$...
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      How does proof assistant organize knowledge?

      I am reading a paper Ittay Weiss, The QED Manifesto after Two Decades — Version 2.0, Journal of Software, 11 no. 8 (2016) pp. 803–815, doi:10.17706/jsw.11.8.803-815 The paper says Goal 7: ...
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      68 views

      Is second order ordinal arithmetic stronger than the multi-sorted first order based variant of it?

      In posting: Is ZFC interpretable in a kind of an extended form of second order arithmetic? I aimed to prove the consistency of ZFC by it being interpretable in what may be better named as "second ...
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      365 views

      Simple book on model theory

      I was expressed by how Mendelson describes models in his Introduction to mathematical logic. Now I am looking for a nice model theory guide. The book (video source, etc.) must: Include the concrete ...
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      321 views

      Is Hilbert's tenth problem undecidable for multilinear polynomials in $\mathbb Z[x_1,\dots,x_n]$?

      A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$. Is there no general purpose algorithm for ...
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      166 views

      Absolute formulas with high complexity

      This is a repost of a MSE question. It is a standard result that $\Sigma_1^{\mathsf{ZF}}$-formulas are upward absolute between $\mathsf{ZF}$ $\in$-models, while $\Pi_1^{\mathsf{ZF}}$-formulas are ...
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      0answers
      141 views

      $\mathcal{C}$-filtering of modules inherited by submodules

      I'll state the question about modules, but I'm open to examples in other contexts. I am not an algebraist, so please forgive any non-conventional terminology. DEFINITION: Let $\mathcal{C}$ be a ...
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      1answer
      217 views

      Uncountable disjoint closed coverings of $[0,1]$

      It is well known that the unit interval $[0,1]$ cannot be decomposed as a countable union of pairwise disjoint closed (nonempty) subsets. See for instance this math.stackexchange question. The proof ...
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      votes
      1answer
      83 views

      On asymptotic classes of finite structures

      I have a question about the following paper: One-dimensional asymptotic classes of finite structures by Macpherson and Steinhorn (link at Trans. AMS website). Let $\mathbf{K}$ be a one-dimensional ...
      3
      votes
      0answers
      155 views

      Can third-order arithmetic prove the consistency of second-order arithmetic?

      I'm trying to get a deeper understanding of Buss's version of G?del's speedup proof. In short, if we assume that $Z_0$ is first-order arithmetic, $Z_1$ is second-order arithmetic, and so on, then for $...
      34
      votes
      2answers
      1k views

      Automatically solving olympiad geometry problems

      Warning: I am only an amateur in the foundations of mathematics. My understanding of this Wikipedia page about Tarski's axiomatization of plane geometry (and especially the discussion about ...

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