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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.

        4
        votes
        0answers
        75 views

        Order types of models of theories of ordinals

        For $C$ a set of ordinals, let $\mathcal{L}(C)$ be the language with identity, a relation symbol for less than, function symbols for successor, addition, multiplication, and exponentiation, and a ...
        0
        votes
        0answers
        23 views

        Proof of consistency of proof system syntactically [on hold]

        I am trying to define semantics of propositional logic using its syntactics. And I am having trouble proving the "only one" part of following problem. Let $A$ be a set of propositional symbols, $\...
        5
        votes
        1answer
        132 views

        Definability of the ring of integer in algebraic extensions of $\mathbb Q$

        J. Robinson has proved that exists a formula $\psi(x)$ in the language of rings which,applied to the rational numbers, defines the the ring integers (making the theory of $\mathbb{Q}$ undecidable, due ...
        0
        votes
        0answers
        32 views

        McNaughton functions and hypersequents for Lukasiewicz logic

        This question was asked at MSE (https://math.stackexchange.com/questions/2951102/mcnaughton-functions-and-hypersequents-for-lukasiewicz-logic) more than half a year ago but there have been no answers ...
        3
        votes
        1answer
        111 views

        Can power set axiom be proved in a class theory of well ordered hereditarily accessible sets?

        I've asked this question at MathStackExchange, only to receive no answer. I'll repost this question here: Working in a pure class theory, where sets are defined as elements of classes. That is: ...
        17
        votes
        4answers
        2k views

        What do we gain with higher order logics?

        G?del's speed up theorems seem to say that higher order logics offer shorter shortest proofs of various propositions in number theory. Explicitly, he gave the following Theorem. Let $n>0$ be ...
        8
        votes
        0answers
        76 views

        Monadic second-order theories of the reals

        I’m looking for a survey of monadic second-order theories of the reals. I’m starting from a 1985 survey by Gurevich which says (p 505) that true arithmetic can be reduced to “the monadic theory of ...
        -4
        votes
        0answers
        64 views

        What form can have statements that are provable by the method of mathematical induction? [closed]

        I am not a professional mathematician, however, I feel that this question is more suitable for MO than MSE. For example, the statement that the sum of the first $n$ natural numbers equals $\dfrac{n(n+...
        1
        vote
        0answers
        80 views

        Sentential, first order and higher logics from a categorical perspective

        Is there a standard reference for understanding sentential, first and higher order logics from a categorical perspective? I'm close to knowing enough $1$/$2$/internal category theory to tackle the ...
        0
        votes
        0answers
        180 views

        Good texts (other than Kunen and Jech) on set theory, specifically on consistency proofs (reflection theorems, absoluteness, etc) [closed]

        I'm finding Kunen and Jech bit of a hard read, and cannot seem to find good alternatives. Please suggest.
        22
        votes
        9answers
        2k views

        Defining the standard model of PA so that a space alien could understand

        First, some context. In one of the comments to an answer to the recent question Why not adopt the constructibility axiom V=L? I was directed to some papers of Nik Weaver at this link, on ...
        4
        votes
        1answer
        302 views

        Turing independent refinement

        Suppose $\kappa< 2^{\aleph_0}$ and $\langle P_i : i < \kappa\rangle$ is a sequence of perfect subsets of $2^{\omega}$. Can we find $Q_i \subseteq P_i$ for $i < \kappa$ such that each $Q_i$ is ...
        13
        votes
        2answers
        443 views

        Is categoricity retained when reducing the language?

        Suppose $\mathcal L \subseteq \mathcal L’$ are first-order languages, $\kappa$ is a cardinal, and $T’$ is a theory in $\mathcal L’$ that is $\kappa$-categorical. Let $T = T’ \restriction \mathcal L$. ...
        18
        votes
        2answers
        2k views

        Status of proof by contradiction and excluded middle throughout the history of mathematics?

        Occasionally I see the claim, that mathematics was constructive before the rise of formal logic and set theory. I'd like to understand the history better. When did proofs by contradiction or by ...
        6
        votes
        1answer
        187 views

        Generic saturation of inner models

        Say that an inner model $M$ of $V$ is generically saturated if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses ...

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