Here are the final drafts of the “Group Process Papers.” Assessment details will be distributed in class. Great work, everyone!
Logic on Math StackExchange
- In the class, there are $30$ students March 29, 2023I am preparing to take an entrance exam for a university in my country, which will happen soon. As part of my preparation, I have been practicing with some sample math tests provided by the university. However, I recently came across a math problem that I could not understand at all. This is the exercise: […]Simebe
- Why do we care about the decidability of a theory? March 29, 2023I'm new to logic theory and i'm stuck with the understanding of decidability of a logical system. As far as I understand, the decidability of a logical system is related to the existence of an effective (in terms of time) procedure to tell whether a given formula can be proved by using the "instruments" of […]InTheSearchForKnowledge
- Free variables in derivation systems March 29, 2023I work relative to a fixed first-order language $L$. As usual I call those formulas which do not contain free variables sentences. There are several derivation systems for first-order logic. For example, in Goldrei's Propositional and Predicate Calculus a derivation of $\psi$ relative to a set $Ax$ of sentences is a list of formulas $\phi_1,....,\phi_n$ […]Nico
- Is every limit computable set in the Ershov hierarchy? March 29, 2023A set is limit computable, or $\Delta^0_2$, if its characteristic function is equal to $\lim_{s\rightarrow\infty}g(x,s)$ for some total computable function $g$. And given a computable ordinal $\alpha$ and a path $P$ through Kleene's $O$, a set is $\alpha$-c.e. if its characteristic function is equal to $\lim_{s\rightarrow\infty}g(x,s)$ for some total computable function $g$ such that there […]Keshav Srinivasan
- Theorems (or references to analysis) of a particular Hilbert-deductive-system using $\\{\neg, \wedge, \vee, \rightarrow\\}$ as primitive symbols? March 29, 2023Context The System CL In section 6.3 of Topoi, Robert Goldblatt describes a Hilbert-style deductive calculus (the only inference law is modus ponens) for the propositional logic of a language with the primitive logical connective symbols $\\{\neg, \wedge, \vee, \rightarrow\\}$, which he calls CL, with following axiom schemata: $\alpha\rightarrow(\alpha\wedge\alpha)$ $(\alpha\wedge\beta)\rightarrow(\beta\wedge\alpha)$ $(\alpha\rightarrow\beta)\rightarrow((\alpha\wedge\gamma)\rightarrow(\beta\wedge\gamma))$ $((\alpha\rightarrow\beta)\wedge(\beta\rightarrow\gamma))\rightarrow(\alpha\rightarrow\gamma)$ $\beta\rightarrow(\alpha\rightarrow\beta)$ $(\alpha\wedge(\alpha\rightarrow\beta))\rightarrow\beta$ $\alpha\rightarrow(\alpha\vee\beta)$ […]Alexander Sanchez
- Is Robinson arithmetic the weakest incompletable system of arithmetic? March 28, 2023Robinson arithmetic (Q) is weaker than PA. We know that any theory that interpret Robinson arithmetic will be incomplete as well. It seems Robinson found his axioms noting what was necessary to conclude the incompleteness proof. But what if there is some very strange axiom system we don't really think that is connected to arithmetic, […]Lost definition
- Can there exist a set $S$ and $F_1, F_2 \in ([5] \times S)^{([5]^S)}$ such that for all $f_1,f_2\in [5]^S$, $F_1(f_1)\in f_2$ or $F_2(f_2)\in f_1$? March 28, 2023Here $[5]:=\{0,1,2,3,4\}$ and for sets $O,I$, $O^I$ denotes the set of functions from $I$ to $O$. The motivation for this question comes from the following axiom of choice "paradox": Let $f_1, f_2 \in [5]^{\mathbb{N}}$ be unknown functions. Player 1 and player 2 define an equivalence relation $\sim$ on $[5]^{\mathbb{N}}$ where two functions are equivalent if […]Terence Coelho
- Subsumption in Polynomial-Time for EL Description Logic extended with non-Nested Negations March 28, 2023Is the subsumption between two formulas of the EL Description Logic extended with non-Nested Negations decidable in polynomial time ? EL extended with non-Nested Negations is defined by : Concept Negation as long as it is NOT nested inside another negation Concept Conjunction Existential Restriction (unlimited) I am only considering the subsumption problem of two […]P. P.
- Counter-example for problem of elementary equivalence? March 28, 2023Show that for systems $\frak{A\subseteq B}$ of a signature $\Sigma$ we have got $\frak{A\prec B}$ if $\Sigma$ contains only one symbol $\sim$ of a two-place relation which is interpreted in $\frak B$ as an equivalence with an infinite number of infinite equivalence classes and the set $A$ (the domain of $\frak A$) contains an infinite […]Maxim Nikitin
- Some questions about the Hyperuniverse Program March 28, 2023The Hyperuniverse Program, founded by Sy D. Friedman, intends to produce new second-order axioms of set theory which appropriately formalize "the universe is maximal" in one of a few ways. A height-maximality principle is one intended to formalize "the class of ordinals is as long as possible" (1, p.9). First-order axioms are not considered, because […]C7X
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