# Tag: perfect circle

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- How do we escape infinite regress when discussing truth in mathematical logic? [duplicate] February 22, 2024In (first-order) logic, I understand that there are two notions of the truth of a sentence $\phi$ in a theory $T$: Syntactic truth: $T\vdash \phi$ if $\phi$ is provable from $T$, Semantic truth: $T\models \phi$ if $\phi$ is true in every model $M$ of $T$. Naturally, both the notion that "$\phi$ is provable from $T$" […]M. Sperling
- Bocce league: How many wins to advance to playoffs? February 22, 2024In our bocce division, we have $14$ teams (including us). Everyone plays each other $1$ time during the $13$ week season ($13$ games for all). At the end of the $13$ weeks, the top $6$ teams advance to the playoffs. What is the minimum wins a team must have to land in the top $6$? […]Jill Keogh
- Example of an uncountable subset of $\mathbb R$ which cannot be proved to have the same cardinality as $\mathbb R$ February 22, 2024I am new to mathematical logic so forgive me if this is a bad question. I understand that the Continuum Hypothesis (CH) is independent of ZFC and therefore there exist models of ZFC in which the CH is false. In such models, by the very definition of CH being false, there must exist a set […]Oliver
- How does one perform induction on integers in both directions? February 21, 2024On a recent assignment, I had a question where I had to prove a certain statement to be true for all $n\in\mathbb{Z}$. The format of my proof looked like this: Statement is true when $n=0$ "Assume statement is true for some $k\in\mathbb{Z}$" Statement must be true for $k+1$ Statement must be true for $k-1$ My […]June Richardson
- Is $F(x) \in z$ absolute February 21, 2024For an absolute Function F (meaning that the formula y=F(x) is absolute), is there a proof that $F(x) \in z$ is an absolute Formula? This Lemma from Kunen applied for $\Phi(w,z)$ being $w \in z$ and $G_1=F$ says there should be: [![Lemma from Kunen][1]][1] However the proof of the lemma is not convincing so i […]Rubids
- Why is having T in every row under two propositions is not sufficient to say that they are equivalent? [duplicate] February 21, 2024Task on propositional logic: "Suppose you create a truth table for A and B, both formulas in propositional calculus, and have a look at the columns below the main connectives of A and B. When do we know for sure that A ≡ B is true?" My understanding is that this option should be correct: […]Роман Кирьянов
- What does $\forall x (Triangle(x) \iff \exists y (Square(y) \land AboveOf(x,y)))$ imply? February 21, 2024I'm trying to make a Tarski World for this structure: $$\forall x (Triangle(x) \iff \exists y (Square(y) \land AboveOf(x,y)))$$ I think that it means the following: Element is a triangle if and only if there is a square below it. So every triangle must have squares below it. There is a square that is below […]Роман Кирьянов
- Irving M. Copi, Logic 14th, 554pg 11 question about error February 21, 2024premise 1 premise 1: (∀x)(Fx→Gx) premise 2: (∃x)(Fx ∧ ~Gx) conclude: (∃x)(Gx∧~Fx) Fa ∧ ~Ga 2. E.I. Fa->Ga 1. U.I. Fa 3. Simp. ~Ga 3. Simp Ga 4, 5. M.P. ~Fa 4, 6. M.T. Ga ∧ ~Fa 7, 8. Conj. (∃x)(Gx∧~Fx) 9. E. G. Q.E.D. The book suggests that this argument is a valid argument. […]김성도
- Parsing this wff efficiently February 21, 2024I have the wff: $$\alpha = (((\neg(A_1\rightarrow (A_3\vee (\neg A_2))))\wedge(A_4\wedge(\neg A_1)))\rightarrow((\neg(A_3\vee A_2))\rightarrow(((\neg A_1)\wedge A_4)\vee A_3)))$$ I've already parsed through $12$ of the $16$ possibilities in the truth table. If we let $$\beta = ((\neg(A_1\rightarrow (A_3\vee (\neg A_2))))\wedge(A_4\wedge(\neg A_1)))$$ and $$\gamma = ((\neg(A_3\vee A_2))\rightarrow(((\neg A_1)\wedge A_4)\vee A_3))$$ Then $\alpha = (\beta \rightarrow \gamma)$. I've already shown […]Cotton Headed Ninnymuggins
- How can mathematical logic try to model math, when mathematics are used to define mathematical logic? February 20, 2024I've done so far a few courses in logic and formal verification, and I've always wondered: mathematical logic, at least as Hilbert envisioned, tries to model mathematics. Formally define what a "true" statement is, or why proving something (at least in a sound system) makes it true. But, every logic course uses mathematics in its […]sadcat_1

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