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        Questions tagged [set-theory]

        forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.

        1
        vote
        1answer
        111 views

        Compare direct limits of two pairwise isomorphic direct systems

        Assume there are two directed systems $(A_i,f_{ij})$ and $(A'_i,f'_{ij})$ of groups over the directed set $(\mathbb{N},\le)$, such that for any $i,j$ there exists isomorphisms $\phi_{ij}:A_i\to A_i'$ ...
        12
        votes
        0answers
        131 views

        Universal locally countable partial order

        Call a poset locally countable if the set of predecessors of every member of the poset is countable. Is the following consistent? There is no locally countable poset $P$ of size continuum such that ...
        0
        votes
        0answers
        158 views

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

        I'm finding Kunen and Jech bit of a hard read, and cannot seem to find good alternatives. Please suggest.
        4
        votes
        1answer
        233 views

        A simple proof for a case where: $\mathbf{L}_\mu \models ZF^-$?

        I am looking for a simple proof (no fine structure, please) of the following: Let $\lambda$ be a limit ordinal, and $\mu < \lambda$, infinite: If $\mathbf{L}_\lambda \models \texttt{"}\mu \mbox{ ...
        21
        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
        281 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 ...
        0
        votes
        0answers
        31 views

        Cardinal exponentiation [migrated]

        My understanding of exponentiation of cardinals leads to the conclusion that if $2 \leq \kappa \leq \lambda$, then $2^\lambda = \kappa^\lambda$ , because: $2^\lambda \leq \kappa^\lambda \leq (2^\...
        7
        votes
        1answer
        237 views

        Exterior powers and choice

        Under the assumption that any vector space has a basis (so under the assumption of the axiom of choice), we can prove the following algebraic statements : 1) If $\varphi:V\to W$ is an injective ...
        6
        votes
        1answer
        186 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 ...
        22
        votes
        7answers
        2k views

        Why not adopt the constructibility axiom $V=L$?

        G?delian incompleteness seems to ruin the idea of mathematics offering absolute certainty and objectivity. But G?del‘s proof gives examples of independent statements that are often remarked as having ...
        12
        votes
        1answer
        923 views

        Existence of a model of ZFC in which the natural numbers are really the natural numbers

        I know that, from compactness theorem, one can prove that there are models of first order arithmetic in which there is some "number" which is not a successor of zero, in the sense that it is strictly ...
        3
        votes
        1answer
        191 views

        Is there a universally meager air space?

        Let $\mathcal P$ be a family of nonempty subsets of a topological space $X$. A subset $D\subset X$ is called $\mathcal P$-dense if for any $P\in\mathcal P$ the intersection $P\cap D$ is not empty. A ...
        2
        votes
        1answer
        117 views

        Injective choice function for “lines” in an infinite cardinal

        Let $\lambda$ be an infinite cardinal and suppose ${\cal L}$ is a collection of subsets of $\lambda$ such that $|k| = \lambda$ for all $k\in {\cal L}$ and, if $k_1\neq k_2\in {\cal L}$ then $|k_1\cap ...
        10
        votes
        1answer
        219 views

        Does the statement 'there exists a first-order theory $T$ with no saturated models' have any set theoretic strength?

        Exercises 6, 7, and 8 in section 10.4 of Hodges' big model theory textbook contain an outline of a proof of the consistency of the following statement There exists a countable first-order theory $T$...
        18
        votes
        2answers
        926 views

        How much Replacement does this axiom provide?

        (There have been many questions on MathOverflow about the axiom scheme of replacement, including a few with a similar flavour to mine. Some have very informative answers and link to excellent papers ...

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