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Results tagged with lo.logic
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user 4137
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.
6
votes
Accepted
Why does the Kleene Hierarchy not collapse?
You're right that the statement $\varphi(a,q,v,w)$ defined by $\forall x<a+q \,\, \exists y<a+v \,\, \forall z<a+w [P(x,y,z)]$ can be checked by a Turing machine. If I read you correctly, you're wond …
- 2,305
4
votes
Accepted
Is there any literature about inner-replacement rule?
I don't know a name for the particular inference you indicate, but its feature that it operates "deeply" within the formulas at hand rather than at the root of their parse trees brings to mind current …
- 2,305
7
votes
Accepted
Does model-complete in a language with a constant symbol imply EQ?
Per JDH's suggestion, I'll turn my earlier comment into an answer.
Assuming $T$ to be model-complete, then whenever $M$, $N$ and $A$ are all models of $T$, it would certainly follow from $A \subset …
- 2,305
3
votes
Is there a general theory of models that has as instances classical FOL, classical propositi...
What you are after is institution-independent model theory, for which Diaconsecu has a recent textbook account. Here the bare concept of an institution is basically that of a logical system absent par …
- 2,305
4
votes
Accepted
Is this system identical to S4.4?
$\mathbf{KT}+\mathbf{AR1}$ is strictly weaker than $\mathbf{KT}+\mathbf{4.4}$. Consider the Kripke frame that is the reflexive closure of the following graph (so that any model built on it is a model …
- 2,305
6
votes
Accepted
Modal logic - satisfiability
Your question as originally written (which Henry correctly diagnosed as problematic in two ways) does not match the more reasonable aim reflected in your comments to Henry's answer. Specifically, you …
- 2,305
7
votes
Accepted
History of Logic Development
The scope of the figures you mention (Tarski, Frege, Peano, Wittgenstein, Russell) makes it a little unclear exactly what you're after. For instance, From Frege to Goedel (as mentioned by Mahmud) is a …
13
votes
Set-Theoretic Issues/Categories
Regarding (1), the definition of category already doesn't rule out instances in which the collection of morphisms between two objects might be class-sized; these are known as categories which fail to …
- 2,305
11
votes
A question about Quine's set theory NF.
One can give stratified definitions for individual Frege-Russell natural numbers, and then so too for the set $\mathbb{N}$ of all Frege-Russell naturals, so that exists in NF. One can then check that …
- 2,305
5
votes
Accepted
Alternate proof of Morley's theorem?
Regarding (2), some evidence that Baldwin refers to some sort of $L_{\omega_1 \omega}$ version of Morley's theorem, rather than just an alternate proof making use of $L_{\omega_1 \omega}$ machinery, c …
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11
votes
Higher categories in logic
You would probably enjoy checking out homotopy type theory and Vladimir Voevodsky's corresponding program of univalent foundations for mathematics. Steve Awodey's survey article (linked to from that s …
- 2,305
4
votes
1
answer
284
views
Who first proved there's an $\omega$-model of $\mathsf{WKL}_0$ in which all sets are low?
I am trying to pin down: who first proved that $\mathsf{WKL}_0$ has an $\omega$-model in which every set is of low degree? As shown in Simpson's Subsystems of Second Order Arithmetic (Theorem IX.2.17) …
- 2,305
8
votes
Accepted
An undergraduate's guide to the foundational theorems of logic
Edit: This answer was given to the original formulation of the question, which asked for five-minute explanations for laypersons met on the street, rather than handwavy introductions for undergraduate …
4
votes
Accepted
Reference Request: Non-Standard Models of PA
Richard Kaye's book Models of Peano Arithmetic is good and accessible. And I know that what Frank said in his comment, about its availability as a pdf online, is indeed true; though like Frank, I shan …
- 2,305
4
votes
effective/constructive/algorithmic probability theory
A recent paper that gives the sort of effective result you're after is Freer and Roy's Computable exchangeable sequences have computable de Finetti measures. From their introduction:
The classical re …
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