Tag: perfect circle
Handy Links
Logic on Math StackExchange
- Some Questions about Godel's First Incompleteness Theorem July 27, 2024Let $\mathfrak{T} = (\mathsf{L}, \mathsf{S},\mathsf{P})$ be a theory for natural numbers, where $\mathsf{L}$ is a first order language for syntax, $\mathsf{S}$ is the corresponding structure (model) for semantics whose universe is natural numbers $\mathbb{N}$, and $\mathsf{P} = (\mathsf{Ax}, \mathsf{D})$ is our proof system. $\mathsf{Ax}$ is the set of axioms containing first order logic axioms and […]Hosein Rahnama
- Two equivalent definitions of function? July 27, 2024I am studying Kenneth Kunen's The Foundations of Mathematics, where the definition of a function is given as $$ \forall x\in\mathrm{dom} (R)\,\exists! y\,[\langle x,y\rangle \in R]\tag{1} $$ I also came across a second definition $$ \forall x\,\forall y\,\forall y'\,[\langle x,y\rangle\land\langle x,y'\rangle \in R]\Rightarrow [y=y']\tag{2} $$ Both definitions seem correct, but I am unsure how to […]user1361001
- What do many-valued logics have to do with independence? July 27, 2024This is a similar question to this post Propositional calculus , axiom scheme independence proof. I've read the first chapter of the book "Introduction to Mathematical Logic" (Mendelson) on propositional logic. Definition of Independence (Def1) according to Mendelson : "A subset $Y$ of the set of axioms of a theory is said to be independent […]Nuno Ricardo Serafim
- Non-termination proof for all non-terminating algorithms? July 26, 2024Let Alg be the set of all algorithms (TM descriptions/ python code, etc.). This set is countably infinite. In classical logic, the following statement is provable: HaltOrNotHalt == ∀ alg ∈ Alg : Halts(alg) ∨ DoesNotHalt(alg) where the predicates are defined as follows: Halts(alg) == ∃step:Nat, RunFor(step, alg) = 'halt' DoesNotHalt(alg) == ∀step:Nat, RunFor(step, alg) […]Suraaj K S
- Question regarding Fitch's paradox of knowability July 26, 2024Section 2 of Fitch's paradox of knowability states the following. Let $K$ be the epistemic operator 'it is known by someone at some time that.’ Let $\diamond$ be the modal operator ‘it is possible that’. Suppose the knowability principle (KP)—that all truths are knowable by somebody at some time: $$\text{(KP)} \qquad\qquad \forall p(p\rightarrow \diamond Kp).$$ […]Hans
- Constructive proofs and terminating algorithms July 26, 2024I was thinking about the following judgement: ¬ ∀n:Nat ¬ϕ(n) ⊢ ∃n:Nat ϕ(n) I am not sure whether this can be proven constructively, because intuitively, it doesn't look like I can find a 'witness' n such that ϕ(n). However, we do seem to have an algorithm that can generate witnesses given the premise, because I […]Suraaj K S
- Help for inventing Hilbert-style deduction system for a language containing many logical connectives and quantifiers July 26, 2024The challenge is the following: Create a formal deduction system consisting of Modus Ponens (MP) as the sole inference rule for a language containing the following symbols: ¬ (not) ∧ (and) ∨ (or) → (implies) ↔ (iff) ← (is implied by) ∃ (there exists) ∀ (for all) ↑ (nand) ↓ (nor) ̸→ (doesn't imply) ̸↔ […]Nuno Ricardo Serafim
- Distribution of Universal Quantifiers when arbitrariness is implied July 26, 2024I want to prove that $B \setminus \cup_{i \in I} A_i ) = \cap_{i \in I} (B \setminus A_i )$ supposing that B is a set, $\{A_i | i \in I \}$ is an indexed family of sets, and $I \neq \emptyset$ . I want to do the proof like that: $$ x \in B […]Alex Pi
- Transitivity of subset relation (proof using hypothetical syllogism). July 25, 2024This question is about a comparatively formal approach to a simple theorem. The theorem says, If $ A\subseteq B$ and $ B\subseteq C$ then, $ A\subseteq C$. Symbolically, $ (A\subseteq B\land B\subseteq C)\implies A\subseteq C$. Most probably we can use the hypothetical syllogism $ (A\implies B\land B\implies C) \implies (A\implies C)$ concept and prove it. […]Mystic mystic
- If $A\Rightarrow B$, how does $\bar{A}$ say nothing about B? [closed] July 25, 2024I'm reading Probability Theory by E.T. Jaynes and in the first chapter on plausible reasoning, he states The proposition $A\Rightarrow B$ does not assert that either A or B is true; it means only that $A\bar{B}$ is false, or, what is the same thing, ($\bar{A}$ + B) is true ... On the other hand, if […]Jon Behnken
Recent Comments