Tag: Open Lab 8
Logic on Math StackExchange
- How would I express these three statements using propositional logic? September 12, 2026Smoking increases the chance of cancer. Smoking does not increase the chance of cancer. Smoking decreases the chance of cancer. Although statements 2 and 3 seem to be negations of statement 1, how would I express these the statements using propositional logic? And how would I show the difference between the latter two statements?user347771
- Can Linear algebra over $\mathbb{R}$ be transferred to real closed fields? September 12, 2026Background: I'm self-studying Real Algebraic Geometry, where the notion of Real Closed Field is introduced. I'm wondering if we can transfer linear algebra (notions, theorems, ...) over $\mathbb{R}$ to any real closed field $R$? For example: Sylvester's law of inertia, for quadratic forms over $R$. The notion of Euclidean space (over $R$) and Hermitian space […]Zoudelong
- Prime-shadow pattern in sums of differences of powers of 3 [closed] September 12, 2026I noticed a curious pattern while exploring powers of 3. Differences between consecutive powers of 3 are always even: 3 1 − 3 0 2 , 3 2 − 3 1 6 , 3 3 − 3 2 18 , 3 4 − 3 3 54 , … If you add consecutive differences, you get: […]voyager
- Is there a metric for "minimality" or minimizing "syntactic complexity" such that we can determine a global minimum bound? September 12, 2026We know that a for a given set of axioms, we can determine the independence of an axiom set: via constructing a model such that no axiom is derivable from any other. I am interested in determining if there exists a global minimal bound for the strength and size of an axiom set. Questions: 1. […]somerandomguy
- Does model theory in set theory require "arithmetization"? September 8, 2026Reading many set theory and model theory textbooks, it's clear that model theory is developed inside set theory, usually ZFC. My question is, does this necessarily require a form of arithmetization? To be clear, by arithmetization I mean not only coding logic symbols as integers, but also as sets more generally. In this sense, my […]conscontreras
- Math shorthand symbol for "assume" September 6, 2026I had a Discrete Methods professor years ago who used a very specific symbol for "assume" that I have never seen anywhere else. It was a capital "S" with a small vertical line through the bottom of it. She would write things like "[symbol] $\varphi$. We have that..." (where $\varphi$ is some formula). She used […]jorb
- Is there a proof of Kolmogorov's theorem using logic? September 3, 2026I believe there are many Kolmogorov theorems, I am referring to the extension one (see https://en.wikipedia.org/wiki/Kolmogorov_extension_theorem ). I encountered it in a course, and the statement feels very compactness-like (in the sense that we have a statement that says finite consistence entails consistence), so I wondered if there is a smart way of using standard […]Tarallino_Birichino
- What are some "everyday" examples of proof techniques? August 31, 2026I once had a teacher who, when introducing "Proof by contradiction", made the observation that although the formal structure could seem daunting on a first encounter, the following argument (in natural language) made sense to everyone: The murder took place in Los Angeles on Saturday, but my client was in New York the entire weekend. […]Robert Lee
- Axiom of choice in the proof that closed sets of Noetherian spaces are unions of finitely many irreducible subsets. [duplicate] August 31, 2026Hartshorne Proposition 1.5 states that any closed subset of a Noetherian topological space can be written as a union of finitely many closed irreducible subsets. A Noetherian topological space is a topological space that satisfies the descending chain condition on its closed subsets. I have one question regarding the proof of this theorem, and another […]khashayar
- Proof-technique for the Following Claim August 26, 2026I am trying to learn how to prove statements by studying proof techniques for common claims. Right now, I am wondering how to prove claims of the following form: Claim. Show that some subset $L$ of the universal set $U$ is the smallest set that contains all sets $S_{1}, \dots ,S_{m}$. For example, the claim […]MickeLil
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