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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.

3,506 questions
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 $\phi_{ij}:A_i\to A_i'$ ...
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 ...
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.
233 views

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 $\mathbf{L}_\lambda \models \texttt{"}\mu \mbox{ ... 9answers 2k views Defining the standard model of PA so that a space alien could understand First, some context. In one of the comments to an answer to the recent question Why not adopt the constructibility axiom V=L? I was directed to some papers of Nik Weaver at this link, on ... 1answer 281 views Turing independent refinement Suppose$\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 ... 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^\...
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 ...
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 ...
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 ...
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 ...
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 ...
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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 $|k_1\cap ... 1answer 219 views Does the statement 'there exists a first-order theory$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\$...
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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