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Results tagged with lo.logic
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user 57583
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.
16
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4
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1k
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Can Suslin (or Aronszajn) lines ever be orderings of abelian groups?
I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way?
Let $\mathcal{C}$ be a class of (torsio …
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11
votes
Forcing as a replacement of induction and diagonal arguments
Cantor's back-and-forth theorem quoted in the OP has a model-theoretic generalization.
If two $\tau$--structures $A$ and $B$ in a vocabulary $\tau$ are partially isomorphic, then there is a forcing e …
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10
votes
Accepted
Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?
The answer is negative apparently. It is consistent relative to ZFC that all Aronszajn trees are
special and that the club principle holds:
http://home.mathematik.uni-freiburg.de/mildenberger/postin …
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10
votes
Accepted
Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper?
It is also stated as Theorem 4.(i), I think, and again on pages 470 and 472 where the reference is given to earlier results.
For a recent easy-to-read presentation of the proof, see Theorem 5.1.4 in …
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9
votes
Sierpinski's construction of a non-measurable set
I think the pages 249-250 are the most relevant source in ariane's pdf. Sierpinski outlines how to go from the cardinality hypothesis to the existence of a non-measurable set, as per Ashutosh's précis …
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5
votes
Use of indiscernibles in model theory
A further application of indiscernibles is to show that a consistent first-order theory with infinite models has models with many automorphisms. In particular, every first-order theory $T$ (in a count …
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4
votes
Accepted
Saturated Ultrapowers
W.W. Comfort, S. Negrepontis, The Theory of Ultrafilters, section 13, in particular Theorem 13.7 and Corollary 13.8, might be useful to you. It contains a textbook presentation of the relevant proofs …
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4
votes
Nice algebraic statements independent from ZF + V=L (constructibility)
One can find examples of algebraic statements that are independent of ZFC + $V = L$ by considering "absolute" versions of standard algebraic propositions. This happens for example when one seeks large …
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4
votes
Accepted
Is it consistent that $\frak{d} < 2^{\aleph_0}$?
Theorem 5.1 in Eric van Douwen's paper "The integers and topology" (Handbook of Set-theoretic Topology) is an old reference for a positive answer to your question and will provide a fuller explanation …
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4
votes
First-order axiomatization of free groups
More can be said than non-first-order axiomatizability. Since the free group $ \mathbb{Z}^{(\omega)}$ is an $L_{\infty, \omega}$-elementary substructure of the non-$\aleph_{2}$-free group $\mathbb{Z} …
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4
votes
Canonical functions in set theory and their applications
In belated partial response to (A), canonical functions are certainly already defined in the Jech-Shelah paper: A note on canonical functions, where reference is made to the work F. Galvin and A. Hajn …
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2
votes
Continuum Hypothesis
A very simple algebraic statement equivalent to CH is the assertion that the Baer-Specker group $\mathbb{Z}^{\omega}$ is almost free, i.e. all its subgroups of cardinality less than $\mid \mathbb{Z}^{ …
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1
vote
Accepted
Diagonalizing against a non stationary set of functions
Sorry this is late and you may already have it all sorted out more attractively. I think an argument might go as follows. It rests on K. Devlin, Variations on $\diamondsuit$, JSL 44 (1979), modified f …
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1
vote
The continuum hypothesis for packing shapes without overlapping
One simple ZFC-observation about $\mathbb{R}^{2}$, along the lines of the finite cross example, is well-known (see for details e.g. P. Komjath, V. Totik, Problems and Theorems in Classical Set Theory) …
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