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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About having one axiom schema for ZFC motivated after the iterative conception of sets?
This posting is related to this posting, and builds its motivation from this answer to it.
Define: $\operatorname {History}(x) \iff \\\forall y \in x: y=\{c \mid \exists z : z \in y \cap x \land (c \...
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Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
I've relatively recently learned about Goodstein's Theorem and its unprovability in Peano arithmetic (the Kirby-Paris Theorem). I do not have any real knowledge of formal logic; but I think I've seen ...
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Algebraization of second-order logic
Is there an algebraization of second-order logic, analogous to Boolean algebras for propositional logic and cylindric and polyadic algebras for first-order logic?
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Rigid non-archimedean real closed fields
Update. The question has been recently answered in the positive by David Marker and Charles Steinhorn (as in indicated in Marker's answer). Note that Remark 3 below is now expanded by reference to a ...
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Bounding proofs of transfinite induction
Let $\phi$ be a "reasonable" formula in the language of first-order arithmetic expressing some amount of transfinite induction along a given (index for a) computable linear order; my default ...
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Defining fields of characteristic zero in existential second-order logic
Is it possible to define in existential second-order logic (ESO) the class of fields of characteristic zero? An easy compactness argument shows that the class of fields of positive characteristic is ...
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Is the field of constructible numbers known to be decidable?
By the field of constructible numbers I mean the union of all finite towers of real quadratic extensions beginning with $\mathbb{Q}$. By decidable I mean the set of first order truths in this field, ...
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Completing half of Hilbert's program: Foundations that are conservative over Peano Arithmetic
The goal of the Hilbert program was to find a complete and consistent formalization of mathematics. Gödel's first incompleteness theorem establishes that completeness is impossible with first-order ...
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Axiomatic strength of the cumulative hierarchy
In the 2021 paper Level Theory Part I: Axiomatizing the Bare Idea of a Cumulative Hierarchy of Sets by Tim Button, a first order theory of the cumulative hierarchy is explored. Initially no axioms ...
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What determines non-finite axiomatizability of a class extension of a set theory?
Suppose $T$ is a set theory, i.e. doesn't have proper classes. And $T$ can interpret $\sf PA$, and $T$ is an effectively generated consistent first order set theory. Now, let $T^+$ be a class theory ...
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Which part(s) of this proof of Goodstein's Theorem are not expressible in Peano arithmetic?
EDIT: Noah Schweber helpfully points out that $\mathsf{ACA}_0$ is a conservative extension of Peano arithmetic in which certain aspects of my proof not expressible in Peano arithmetic are expressible. ...
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Algebraic structures on spaces of ultrafilters
The space of ultrafilters on $\omega$ has a natural semigroup structure, and ultrafilters that are idempotent in that algebra have seen applications in combinatorics on the natural numbers, for ...
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Intuition for the "internal logic" of a cotopos
Let $\mathcal{E}$ be an elementary topos. By definition, $\mathcal{E}$ is a category that has finite limits, is Cartesian closed, and has a subobject classifier $\Omega$. This subobject classifier can ...
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Variable-centric logical foundation of calculus
Since calculus originated long before our modern function concept, much of our language of calculus still focuses on variables and their interrelationships rather than explicitly on functions. For ...
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Computational view of subsystems of second-order arithmetic
If System T "corresponds" to full first-order arithmetic, and System F (λ2) corresponds to full second-order arithmetic, what type systems would be associated with weaker fragments, ...
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