Written work – None.
WeBWorK – Assignment #5, due Tuesday, October 9th, at midnight.
OpenLab – OpenLab #4, due Thursday, October 11th, before class.
Logic on Math StackExchange
- Generalizing Bertrand Russell's "Pope" proof: Can any false premise $P$ lead to an arbitrary conclusion $Q$ via a "bridge" February 17, 2026Recently, I came across a famous anecdote regarding Bertrand Russell and the principle of Ex Falso Sequitur Quodlibet. As the story goes, a student challenged him: "If $0=1$, prove that you are the Pope." Russell supposedly replied: "If $0=1$, then by adding $1$ to both sides, $1=2$. The Pope and I are two people. Since […]Humberto José Bortolossi
- Can an infinite field have an $\omega$-categorical theory? February 16, 2026Let $F$ be a field and let $T$ be its complete, first order theory in the language of rings. I'm pretty sure $T$ can never be $\omega$-categorical, but I have not been able to find a proof. If char$(F)=0$, this is trivial, as $tp(0),tp(1),tp(2),tp(3), \dots$ are distinct and so $S_1(T)$ is infinite. But in positive […]Susana Santoyo
- Is the fixed point provided by Kleene's recursion theorem computable? February 16, 2026So Kleene's Recursion theorem states: Given a computable total function $f:\mathbb{N}\to \mathbb{N}$, there is some $e\in \mathbb{N}$ such that $\varphi_e=\varphi_{f(e)}$. where $\varphi_n$ denotes the $n$th computable function. My question is as follows: given a total computable function $g:\mathbb N^2 \to \mathbb N$ and some $y\in \mathbb{N}$, is the function that maps $n$ to the fixed […]4u9ust
- First-order theories interpreting $Q$ beyond arithmetic and set theory [closed] February 15, 2026I'm currently working on a survey of recursively axiomatizable first-order theories that are capable of interpreting $Q$ (basic arithmetic), with an emphasis on their consistency strengths. So far, I have analyzed common subsystems of $\mathrm{PA}$ and $\mathrm{Z}_2$, set theories such as Zermelo set theory, its extensions and $NF$, as well as $ETCS$. I would appreciate […]Disp_Librar1
- Is this a counterexample to the theorem that every admissible rule in classical propositional logic is derivable? February 15, 2026I read somewhere that, in classical propositional logic, it is a theorem that every admissible rule is derivable. However, I have thought of an apparent counterexample to that theorem. First, let our propositional language consist of a countably infinite set of propositional variables $\{P,Q,R,P_1,Q_1,R_1,...\}$, along with the negation symbol $\neg$ and the conditional symbol $\rightarrow$, […]user107952
- $(\tau \setminus \{\emptyset\}, \subseteq^{*})$ is not separative February 14, 2026Let $(X,\tau)$ be a topological space. We write $A \subseteq^{*} B$ if $A \cap B$ is dense in $A$ (equivalently if $Reg(A) \subseteq Reg(B)$, where $Reg(..) := Int(Cl(..))$). It is straightforward seeing that $(\tau \setminus \{\emptyset\}, \subseteq^{*})$ defines a qoset (quasi ordered set). Given a qoset $(Q,\leq)$ we say that it is separative if for […]user1570557
- Writing a result in logic February 13, 2026I would like to see if I am translating correctly from natural language into logic with symbols. Let's say I have the following result: Let $m,n\in\mathbb{N}$, $A\in\mathbb{Q}^{m\times n}$, $b\in\mathbb{Q}^m$ with $b\neq\mathbf{0}_m$ be such that $Ax=b$ has a binary solution in $x$. Then there exists a function $f_{A,b}$ from $\{0,1\}^n$ onto itself such that the least […]Marc Dinh
- Showing the induced order on a Hilbert algebra is transitive February 13, 2026A H-algebra is an algebra $(A, \rightarrow, 1)$ of type $(2,0)$ such that the following axioms are fulfilled for every $\alpha, \beta, \gamma \in A$ : $$ \begin{aligned} & \left(a_1\right) \alpha \rightarrow(\beta \rightarrow \alpha)=1 ; \\ & \left(a_2\right)(\alpha \rightarrow(\beta \rightarrow \gamma)) \rightarrow((\alpha \rightarrow \beta) \rightarrow(\alpha \rightarrow \gamma))=1 ; \\ & \left(a_3\right) \text { If } […]NiceJJ
- Is there a consistent set of axioms sufficient to analyze basic algorithms (such as merge sort)? February 12, 2026I will focus on merge sort to make the question specific and answerable, but the spirit of my question is to allow replacing merge sort with any basic algorithm (an ill-defined term, hence the focus on merge sort). Is there a set of axioms, known to be consistent, where we can prove that: Merge sort […]Sebastian
- Are two continuum-sized structures that are "the same at all sub-continuum levels" necessarily isomorphic? (Counterexample in ZFC) February 9, 2026Now cross-posted to MO. This is a follow up to my previous question: Let $\mathcal{L}$ be a countable language and $M, N$ be two structures over $\mathcal{L}$. Suppose $|M| = |N| = 2^{\aleph_0}$ and they furthermore satisfy the following condition: For each $m \in M$ there is a countable subset $N_m \subseteq N$ and for […]David Gao
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