Tag: Wau
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- Are two continuum-sized structures that are "the same at all sub-continuum levels" necessarily isomorphic? (Counterexample in ZFC) February 9, 2026This is a follow up to my previous question: Let $\mathcal{L}$ be a countable language and $M, N$ be two structures over $\mathcal{L}$. Suppose $|M| = |N| = 2^{\aleph_0}$ and they furthermore satisfy the following condition: For each $m \in M$ there is a countable subset $N_m \subseteq N$ and for each $n \in N$ […]David Gao
- Is any proof of $P(x)$ for incomputable number $x$ valid also for some other values of $x$? February 8, 2026Let $a$ be a non-computable real number. Suppose there is some property $P(x)$ for which $P(a)$ is provably true (within some consistent axiomatic system, say ZFC). Does that mean that this proof is valid not only for $a$, but for some strictly larger set of numbers? The intuition behind this question is that since $a$ […]kvardekkvar
- My problem with Russel paradox [closed] February 8, 2026My intuitive approach. Problem is how are defined elements of the set and question is which set. Because there is Use–mention distinction Set A and set "A". And if you ask about question membership of an element in a set in the sense of metalogic.which set A or "A" In a sense Russell's paradox is […]someone
- I can't figure out how to directly prove cut-elimination for Herbrand style proof system February 7, 2026I should clarify the proof system I am talking about. In this system we prove things in the following way: given a set $\{\Gamma_1,\Gamma_2,...\}$ of closed statements in first order logic, $\Gamma$ is implied by these statements if the following holds: "Let $\{P_1,P_2,...\}$ be predicates appearing in any of the $\{\Gamma_1,\Gamma_2,...\}$ or in $\Gamma$, let […]Pineapple Fish
- Translate "Some fragile items are transparent only if they are glass" February 6, 2026What is the predicate expression for statement : "Some fragile items are transparent only if they are glass." Let: $Fx = x$ is fragile item $Tx = x$ is transparent item $Gx = x$ is glass item Does this mean $(Fx\wedge(Tx\rightarrow Gx))$ or $(Fx\wedge Tx\rightarrow Gx)$ and why? Is it because transparent is in the […]user14271528
- How to construct this Kripke–Platek set theory with urelements (KPU) model that has nonstandard natural numbers February 5, 2026Suppose that I have a model $\mathfrak{A}$ that satisfies a set of first-order axioms (i.e. KPU Axioms). KPU is stronger than PA. I read from Admissible Sets and Structures: An Approach to Definability Theory by Jon Barwise that by ordinary compactness, there exists an elementary extension $\mathfrak{N}$ of $\mathfrak{A}$ that contains non-standard natural numbers. But […]Link L
- Proving the link between a valid biconditional and logical equivalence in first-order logic February 4, 2026What prompted this question was this comment I saw by Mauro ALLEGRANZA, who wrote: Basically, the biconditional is a connective (a part of the language) while logical equivalence is a semantical relation between formulas expressible in the meta-language. The two are strictly linked: $\models (\varphi {\iff} \psi)$ if and only if $(\varphi \models \psi)$ and […]Anthony
- case definition of recursive functions February 3, 2026Let $f_1,..,f_n: \mathbb{N} \to \mathbb{N}$ recursive functions and let $\langle R_i : i=1,..,n\rangle$ be a recursive partition of $\mathbb{N}$. Now let $g: \mathbb{N} \to \mathbb{N}$ be defined as: $g(x) = \sum_{i=1}^{n} f_i(x) \cdot \chi_i(x)$ Is it true that $g(x)$ is still recursive? The reason why I'm asking is that I'm trying to prove the following […]user1570557
- What are the formal definitions of admissible rule versus derivable rule? [closed] February 3, 2026I am having a hard time understanding the distinction between admissible rules and derivable rules in logic. I don't see how the concepts are different. I guess I could understand the distinction better if the definitions were formalized. Can someone give me a text where there are formal definitions of both admissible rules and derivable […]user107952
- Could we use paraconsistent logic to create a system where 1/0 'works'? [closed] February 2, 2026I am aware of systems that try to define 1/0, like wheels, but I was wondering if we could take another approach. If I remember correctly, those other systems try to make division by zero consistent, but often end up bending arithmetic rules, and I was wondering if perhaps we should go a different direction […]Detective Kid
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