Written work: NOTE: The following suggested problems are for practice only, and will NOT be collected.
Section 11.4 p194: 2, 3, 5, 6, 7
Handout: Theorem NT 6.2, 6.3
WeBWorK – none
OpenLab – none
Project Reflection – Due before the final exam, Thursday 12/20.
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Logic on Math StackExchange
- Categoriсal perspective on the Disjuntion property of the intuitionistic propositional calculus June 1, 2024I came about four different proofs of the disjunction property: formulated in the language of Heyting algebras; done using Kripke models; using the fact that every topological space is an open subspace of a strongly compact space (this can be found in "The Mathematics of Metamathematics" by Helen Rasiowa and Roman Sikorski, IX.6 p.392). These […]Georgii
- Understanding Proof of Lemma 5.6.8 of Mathematical Logic by Chiswell & Hodges June 1, 2024In Mathematical Logic by Chiswell and Hodges, there is part (a) of the lemma below. My question is on Case 1, subcase (iii) in the proof. The authors say $t$ is a variable not among $x_1,...x_n$, and then that $t[T]_A = t_A = t[S]_A$. But... isn't $t_A$ NOT defined in this subcase? By Definition 5.6.2, […]Artyom Elessar
- Is intuitionistic logic a subsystem of classical logic? June 1, 2024Joan Moschovakis' Intuitionistic Logic claims: "Although intuitionistic analysis conflicts with classical analysis, intuitionistic Heyting arithmetic is a subsystem of classical Peano arithmetic. It follows that intuitionistic propositional logic is a proper subsystem of classical propositional logic, and pure intuitionistic predicate logic is a proper subsystem of pure classical predicate logic." What is the meaning of […]shea
- Trouble with combinatorial geometry June 1, 2024Suppose that the figure at right consists of thirty rods of equal length that form twelve pentagonal figures of equal size, which form the twelve sides of a regular dodecahedron. If any two rods are painted white and the remaining twenty-eight are painted black, how many distinct patterns are possible? ImageAlexandra Low
- XOR sum of array June 1, 2024When you are given an array of even number of elements: [$a_1$ $a_2$ $a_3$ ….. $a_n$] ($n$ is even) Assume the $a_i$ are not all zero Let $S$ = the XOR sum of all these original elements You will always be able to remove an element such that the XOR sum of the resulting array […]Ashishkabaab
- Why is the Implication Sign Not Viewed as a Causal Relation in Math? [closed] June 1, 2024Why is the Implication Sign not viewed as a causal relation in math? Now, when I use the word "causal" I mean it in the Aristotelian sense to include efficient causality but to be not just limited to efficient causality. It also includes material, formal, final, and an array of other causes.AUTIST INC
- Smalleness condition of sets May 31, 2024In Hovey's book (Model Categories) he define a small object, and then he proves that every set $A$ is $|A|$-small, but I didn't understand what he did. If $a\in A$, then $f(a)\in colim_{\betaDiego
- Violations of self-identity May 31, 2024Is it possible to have an instance of a mathematical object $x$, or an object in some logical theory, such that there is no way of defining an equivalence relation $R$ for which $R(x,x)$? E.g first order logic without equality is not the kind of example I’m looking for since, though it lacks equality, we […]Joa
- Confused about Cooper's book's proof of Selman's theorem on enumeration reducibility May 30, 2024Here is an excerpt of Barry Cooper's book Computability Theory (2004), bottom half of page 179, which is short enough that I can copy it verbatim: THEOREM 11.1.13 (Selman's Theorem, 1971) For any $A,B\subseteq\mathbb{N}$ $$A \mathrel{\leq_e} B \;\Longleftrightarrow\; \forall X[B\text{ c.e. in }X \Rightarrow A\text{ c.e. in }X]$$ PROOF (sketch) The left-to-right implication I will leave to you. Conversely, assume […]Gro-Tsen
- Why does Gödel go through so many steps in his ontological proof? [closed] May 30, 2024Wouldn't it be enough to define "godlike" any $x$ such that $\forall\varphi(P(\varphi)\implies\varphi(x))$, then define necessary existence $E(x)$ and eventually stating the axiom $P(E)$? Why wouldn't this be enough for the proof? There are many intermediate steps which I don't quite understand the point of.Elvis
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