Tag: grading policy
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- Number of countable models of DLOWE with an increasing order-isomorphism. December 3, 2024Recently I tried to answer this question by taking the DLOWE and an ascending sequence of constants ($c_1 < c_2 < c_3 \cdots$) construction and encoding it in a finite language with a finite axiomatization. If we think of our DLOWE as $(\mathbb{Q}, \le)$, then the sequence of constants converges to a rational, converges to […]Greg Nisbet
- Reference Request: Second Order ZFC December 3, 2024I've seen a fair bit of discussion about second order ZFC (Second order ZFC, intuition required, How is second-order ZFC defined?, A question about Second-Order ZF and the Axiom of Choice, A question about Second-Order ZFC and the Continuum Hypothesis) but I can't find anything about it in my books. Are there good books which […]daRoyalCacti
- On the Definition of Simultaneous Substitution of Terms in FOL formula December 3, 2024Consider a firsr-order language $L$. Let $\varphi$ be a formula, $t$ a term and $x$ is a variable. We define the formula $\varphi[t/x]$, which denotes the formula obtained by substituting $t$ in place of $x$, as follows (here, $P$ is a $n$-ary predicate symbol and $\dagger$ denoted undefined; it does not belong to $L$). $$\varphi[t/x]= […]Ray
- Is there a realistic threshold for formula complexity which might indicate consistency? December 3, 2024Many arguments in favour of the consistency of $\mathsf{ZFC}$ and similar systema point to their empirical usage throughout the years without the apparition of any purported inconsistencies. While I agree wholly with such observations, I'm also left to wonder: could that be because, realistically speaking, we only ever utilize a minute fraction of $\mathsf{ZFC}$ during […]Sho
- Solving theorem. How to prove $\neg(p \land q) \to (\neg p \lor \neg q)$ with lean [closed] December 2, 2024I am trying to make a proof by case for this proposition. So far anything I have tried I am only able to prove it one case and not the other. intro hnotp_and_q by_cases first : p ∧ q . exfalso apply hnotp_and_q apply first In this attempt I can prove when p and q […]Mello
- Example of a (true) statement in PA which cannot be proved in PA and cannot be proved in ZFC? December 2, 2024I'm aware of a number of statements (for example, Con(PA), and those given by Goodstein's theorem and the Paris–Harrington theorem) which are stateable in Peano Arithmetic (PA), but not provable within PA. Each of these is provable in ZFC. Is there an example of a statement in PA which is not provable both in PA […]Ollie Taylor
- How can I ensure that $x \iff p \land q$ here? December 2, 2024Say I am writing a Boolean formula in which only $\land$ is between clauses, and only $\oplus$ is between literals in the clauses. How can I make it such that an assignment of variables satisfies the formula only if, for variables $x,p,q$, $ \ \ x \iff p \land q$? My problem is that I'm […]Princess Mia
- formalize "there exists a function f such that f(a)=b" into first-order formula [closed] December 1, 2024I want to know how to formalize the sentence "there exists a function f such that f(a)=b" into a first-order formula. It should not be $\exists f f(a)=b$ because first-order logic does not allow quantifying over functions. Update: Maybe we need to restrict the first-order language to that for ZFC which has no other relation […]peter
- Problem characterizing Lindenbaum algebras December 1, 2024Given a first order type $\tau$, we use $S^\tau$ to denote the set of $\tau$-sentences, i.e. formulas with no free variables. Let $T = (\Sigma, \tau)$ a theory, with $\Sigma$ the set of axioms in the theory. Then we can define \begin{equation*} \varphi \dashv \vdash_T \psi \iff T \vdash \left( \varphi \leftrightarrow \psi \right) \end{equation*} […]lafinur
- Completeness for existential second-order logic November 30, 2024I find in this link that existential second-order logic is compact. On the other hand, I find in this link that existential second-order logic is not complete. Here is a quote: A "naturally occurring logic" that also serves to show that (2) (compactness) does not imply (1) (completeness) is the existential fragment of second order […]user1868607
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