Here are the final drafts of the “Group Process Papers.” Assessment details will be distributed in class. Great work, everyone!

# Tag: final papers

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- Is this a valid proof of Lemma 17.2 in Boolos' Computability & Logic? November 1, 2024Silly question. I have decided to reopen Boole's Computability and Logic and I'm wondering if there is a (very slightly) different way of proving Lemma 17.2, which states $\def\bg{\textbf{g}}$ 17.2 Lemma. Let $T$ be a consistent theory extending $\mathbb{Q}$. Then the set of Gödel numbers of theorems in $T$ is not definable in $T$. The […]Sam
- Need help understanding the proof of Szemeredi's lemma using Los's theorem October 31, 2024I've found these lecture notes: https://www.math.ucla.edu/~chernikov/teaching/16F-MATH223M/CombinatoricsNotes2.pdf I'm trying to understand the proof of theorem 2.1. (regularity lemma), but it seems like the entirety of the proof (on page 12) is handwaved away. I understand the general idea, but the details elude me. We assume that the theorem does not hold, hence we can find, for […]tses
- Exclusive Or logical equivalences October 31, 2024On the Wikipedia page for Exclusive or, there are logical equivalences given which state: \begin{align} p\not\leftrightarrow q\equiv&(p\land\lnot q)\lor(\lnot p\land q)\\ \equiv&((p\land\lnot q) ∨ ¬p) \land ((p\land\lnot q) \lor q) \end{align} I was able to prove that these are equivalent with a truth table, but I'm trying to figure out which rule of logic causes the […]Tedd Firth
- How does the given statement restrict $x$ to just one member of the universe? October 31, 2024This question is from my discrete mathematics exam: Let the universe comprise all guests at a party. We define the following open formulae: $B(x)$: guest $x$ came to the party by bike $C(x)$: guest $x$ came to the party by car Using $x \ne y$ to mean "$x$ and $y$ are different elements of the […]jules
- Partial completeness of second order logic October 31, 2024It is well-known that First Order Logic is complete, i.e. given a first order theory, every sentence which is true in every model of this theory, can be deduced from the theory, and it is similarly well known that Second Order Logic is not in general complete. However, one can still consider the deductive closure […]Carlyle
- What exactly is the extension of a binary relation? October 30, 2024I have a serious doubt about what is the extension of a relation. The question If I have a binary relation $R \subset A\times B$, it is often said that the extension of $R$ is a subset of ordered pairs $(a,b)$ such that $R(a,b)$ is true, and where $a\in A$ and $b\in B$; but isn't […]Giovanni Piñerez
- There is a largest rational number q whose square is less than 2 October 30, 2024I am stuck on this question. We are currently studying logic in math. This is the task: Write the following statements using quantifiers and indicate the negation of the statements, in quantifier notation and in words. Make sure to use clear brackets. For the negation, it is, of course, not enough to simply put the […]434dx
- What is the constant in the $\forall$I-rule and $\exists$E-rule in Natural Deduction? [duplicate] October 30, 2024In the $\forall$I-rule: $$\frac {\phi(a)} {\forall x \phi[x/a]}$$ and the $\exists$E-rule: $$\frac {\exists x\phi(x)} {\phi[a/x]}$$ the $a$ is said to be an arbitrary constant. I wonder if the FOL(first-order language) does not have a constant, what is the $a$? Even the FOL has constants, because we cannot use a specific constant, $a$ should not be […]William
- Is $\exists x (x=c\land \forall y(y=c\rightarrow y=x))$ a theorem in FOL with constant $c$? [closed] October 30, 2024I wonder for every FOL(first-order logic) with constant $c$, if the following formula is a theorem in it: $$\exists x (x=c\land \forall y(y=c\rightarrow y=x))$$ If the answer is yes, please provide a proof.William
- Knights and Knaves fill in the Chart October 29, 2024I would love some help on this question. We were not taught how to do this so I'm super confused and would greatly appreciate help! Thanks in advance.Violet Fields

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