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Results tagged with lo.logic
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user 4600
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.
2
votes
Accepted
What is ($\Pi^1_1$-CA)${}+{}$BI? And what is ID${}_\omega$?
BI = Bar Induction.
See https://www.jstor.org/stable/2270902 for the abbreviation
and
https://en.wikipedia.org/wiki/Bar_induction for the definition.
- 24.8k
3
votes
Can we have a theory $T$ that is complete for simple sentences in the language of $T$ that a...
Partial answer:
If there is such a theory $T$, it would mean that every simple sentence that's independent of $T$, must decide Con($T$).
Note that every $\Pi^0_1$ sentence is equivalent to a simple …
- 24.8k
32
votes
Accepted
On sentences true in all finite groups
The answer is Yes for the second question, about $(\exists x)(\forall y)w=1$. Following Christian Remling's idea:
If a sentence like
$$\exists x(\forall y)(yxy^{-1}x^2y^{-9}\dots=1)$$
holds in all fin …
- 24.8k
7
votes
Accepted
Is DNC/DNR stronger than "prompt" non-computability?
The graph of the course-of-values variant
$$\{(x,(f(0),\dots,f(x))): x\in \mathbb N\}$$
of such a function would be effectively immune. Namely, if we enumerate a subset of this graph then there is an …
- 24.8k
1
vote
Turing degree of finding independent formulas
From your use of "$\downarrow$" it seems you do not require $f$ to give an answer on non-$\omega$-consistent theories.
At least we can get a function $f$ partial computable in $0^{(3)}$ (perhaps you …
- 24.8k
4
votes
Accepted
On fast-growing hierarchy
Let $\varphi_a $ be the $ a $ th partial computable function in a standard way.
Given a recursively enumerable set $ W $, let $ f (n) $ be the maximum of $\varphi_a (b)$ over $ b\le n $ and $ a $ amo …
- 24.8k
5
votes
Relation between Turing degrees and functions computable with them
Your condition is equivalent to $A''\le_T B'$, that is $B$ is high above $A$.
Here $'$ is the Turing jump operator, i.e., the relativized halting problem operator.
In the case $A=0$, $B$ is high. The …
- 24.8k
1
vote
Is there a good list of nomenclature for modal axioms?
Apparently $\Box\Box\alpha\rightarrow\Box\alpha$ is known as C4 (Converse of (4)).
Source: Stanford Encyclopedia of Philosophy.
- 24.8k
5
votes
${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$
${\frak b}'=\aleph_0$ since you can take $S$ to be the collection of constant functions.
${\frak d}' = {\frak d} + \aleph_0$ since you can take any family $S$ realizing $\frak d$ and close it under f …
- 24.8k
3
votes
Literature on Kripke models
Brian F. Chellas: Modal Logic: An Introduction, 1980.
Starts very basic but covers in detail the beautiful completeness theorem proofs for the basic systems like $S5$, $S4$, $K$.
- 24.8k
18
votes
In what sense is GCD an extension of boolean OR?
In the ordering $\preceq$ of nonnegative integers by divisibility, 1 is the least element and 0 is the greatest, and we have for instance
$$
1\preceq 2\preceq 6\preceq 12\preceq\dots\preceq 0.$$
In th …
- 24.8k
1
vote
Accepted
Logic Alphabet for more than Two Variables
Well, when the number of Boolean variables is $n=3$, since the number of connectives is $2^{2^n}=(2^{2^2})^{n-1}$, we can use infix notation like $pxqyr$, where $p$, $q$, $r$ are Boolean variables and …
- 24.8k
3
votes
Accepted
Jump of strongly hyperhyperimmune degrees and DNR relative to 0'
Carl Jockusch and Frank Stephan. A cohesive set which is not high. Mathematical Logic Quarterly, 39:515-530, 1993. A corrective note (keeping the main results intact) appeared in the same journal, 43: …
- 24.8k
3
votes
Models for the given FOL statement
Yes. If there is a model $\mathcal M$ of size $n$ of any sentence $\phi$ that does not use = then you can take any element $a$ of $\mathcal M$ (using the fact that $n>0$) and let $\mathcal N$ be $\mat …
- 24.8k
21
votes
Accepted
Question arising from Voevodsky's talk on inconsistency
Let $S$ be a first order definable Martin-Löf random set such as Chaitin's $\Omega$. If Peano Arithmetic, or ZFC, or any other theory with a computable set of axioms, proves infinitely many facts of t …
- 24.8k