# Tag: perfect circle

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- the difference of explicit and implicit definition of functions logically June 13, 2024The fundamental analysis book I'm reading has a section on the difference between the implicit and explicit definitions of functions. The implicit definition of a function f specifies what property $ P(x,y) $ links the input x with the output $f(x)$. How is the implicit definition different from an explicit one, showing how one generates […]roro
- Difference of ≡ and ⇔ [duplicate] June 13, 2024I'm studying cs in germany, I have been going through my script starting with propositional logic. I don't really understand the difference between those. I watched some YT-Videos on the topic but i didn't really helped me differentiate between those. They just sometimes use one or the other. "Definition" in my own words (my understanding): […]user1335232
- Morley Rank of disjunction is equal to the maximum of the Morley Rank of the disjuncts June 13, 2024I want to prove that if $X_1, X_2$ are definable subsets (in some suitable theory), then the Morley rank $$\text{MR}(X_1 \cup X_2) = \max(\text{MR(}X_1), \text{MR}(X_2))$$ where MR(-) denotes the Morley rank. This is in Marker's Model Theory (Lemma 6.2.7 ii). I'm trying to prove this by induction and it seems clear to me that this […]XavierbutwithaJ
- What statement is true for every $a$, $b$ and $c$? June 13, 2024I have this task in mathematical logic for which I don't really have a tool for solving. What statement is true for every a, b and c? $a \in b \wedge b \in c \rightarrow a \in c$ $a \in b \wedge b \subseteq c \rightarrow a \in c$ $a \subseteq b \wedge b \in […]Danilo Jonić
- Ramsey's Theorem and Weihrauch reducibility June 13, 2024Let $\text{RT}^n_k$ denote (infinite) Ramsey's theorem for $n$-tuples and $k$ colors. Let $\leq_W$ denote Weihrauch (i.e., uniform) reducibility. It is known that, for fixed $k \geq 2$, if $n > m \geq 2$, then $\text{RT}^m_k \lneq_W \text{RT}^n_k$ (pages 5 and 6 of this paper). It is also known that, for fixed $n \geq 2$, if […]Gavin Dooley
- Defining relations. June 13, 2024Do these answers provide what is required? Even(x)=∃k(x=2*k) Triple(y,x)=∃x(y=3*x)Lior
- Why did Tao include the reflexive axiom for equality here? (Analysis I) June 13, 2024In Analysis I, Tao lists several axioms which equality, defined upon a class of objects $T$, must satisfy. The reflexive axiom he gives is Given any object $x$, we have $x=x$. However, from Wikipedia here, equality between 2 expressions asserts "that the expressions represent the same mathematical object". This makes the reflexive axiom seem superfluous- […]Princess Mia
- Would this logic be considered constructive? June 13, 2024I have asked about similar logics before, but this one is different. The logics that I’ve asked about in the past take the Gödel-McKinsey-Tarski translation for Intuitionistic Propositional Logic to classical $S4$, but change the translation of negation to $t(\neg A)=\neg \Box t(A)$. If you define the translation thus: $t(p)=\Box p$ $t(\neg A)=\neg \Box t(A)$ […]PW_246
- Whats the solution to this mathematical reasoning problem? [closed] June 12, 2024A multiple-choice test question offered the following four options relating to a certain statement: A The statement is true if and only if x > 1 B The statement is true if x > 1 C The statement is true if and only if x > 2 D The statement is true if x >2 […]Saqlain Syed
- Does Dan Willard demonstrate that classical logics with the Law of the Excluded Middle versus those with Double Negation Elimination are distinct? June 12, 2024Context: Dan Willard's 2020 review paper of his work on Self-Verifying Theories/Self-Justifying Axiom Systems (SJAS) is titled "How the Law of Excluded Middle Pertains to the Second Incompleteness Theorem and its Boundary-Case Exceptions" [0]. In it, he claims that by disallowing the Law of the Excluded Middle (LEM) as a axiom schema, SJAS of sufficiently […]jpt4

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