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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.
19
votes
Essential reads in the philosophy of mathematics and set theory
Benacerraf and Putnam's Philosophy of Mathematics: Selected Readings is a pretty standard (as these things go) collection of seminal papers in the philosophy of mathematics generally, and in the philo …
13
votes
How much of ZFC does Quine's New Foundations prove?
NF does prove Cantor's theorem in the sense you indicate, $|\mathscr{P}_1(X)|<|\mathscr{P}(X)|$ for any set $X$. The usual ZF proof goes through, because definitions in that proof which need to be st …
- 2,305
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 …
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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 …
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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 …
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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 …
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 …
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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 …
7
votes
Von Neumann's consistency proof
I'm late to the party, but hope that the following might usefully complement Ali's nice answer and references, first by elaborating slightly on the historical side, and second by briefly indicating an …
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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 …
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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 …
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6
votes
What did Zermelo say he was hoping for on the consistency of set theory?
The following remarks may not speak to what you're really after, but given your explicit reference to Hilbert's Program still being years away when wondering what Zermelo might have in mind, they may …
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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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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
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 …
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