Tag: ViHart
Logic on Math StackExchange
- Are valid, proven theorems the same, semantically, as definitions? Or axioms? December 10, 2025Okay, I know about this question and have been looking for others that might speak directly to this. When I ask this question to Google AI (I don't use AI much at all), Are all valid proven theorems the same as definitions? the answer is very close to what I would predict: No, proven theorems […]robert bristow-johnson
- Why can't ZFC prove an internal version of ZFC to be consistent? December 8, 2025I'm sure everyone is tired of questions that sound like this one, however I have been unable to find an equivalent one asked on here before, so forgive me. Of course, per Godel's second incompleteness theorem, Godel's sentence which I will state simplistically as 'I (ZFC) am consistent', phrased arithmetically in terms of Godel numberings, […]C V Astley
- Cardinality of projective sets under large cardinals December 8, 2025Theorem 12 of these notes on infinite games by Löwe says under $\mathsf{ZFC}+$"there are infinitely many Woodin cardinals and a measurable cardinal above them", the cardinality of a projective set $A$ is either $2^{\aleph_0}$ or $n901
- Are there any applications for this approach to the NOT operator? [closed] December 6, 2025From my understanding of logical operators, NOT can be written as: $$1-x\\x\in\mathbb{S}$$I think I found another way to write it, taking advantage of the presence of zero divisors in sedenions $\mathbb{S}$. $$NOT=(z_1^x\cdot z_2)^4\cdot 4^{-1}\\z_1,z_2\in\mathbb{S}\\z_1=e_6+e_{13}, \;z_2=e_7+e_{12}\\z_1z_2=0, z_1^0=1\;(identity)\\z_2^4=4\because z_2^2=(e_6+e_{13})^2=e_6^2+e_6e_{13}+e_{13}e_6+e_{13}^2\\ =-1+e_{11}-e_{11}-1=-2\\ \Rightarrow z_1^1\cdot z_2=0,\; z_1^0\cdot z_2=z_2$$ This representation seems computationally impractical, but I'm curious if it has any […]Mikkyele
- Generated structures with same diagram must be isomorphic December 5, 2025I'm reading the proof in the Appendix of the paper "Quantifier-free Interpolation of a Theory of Arrays". At the end of the right to left implication, I arrive to the following scenario: I have two structures $N_1$ and $N_2$ generated over a set of constant $C$ on a signature $\Sigma$. I know that the diagram […]user1868607
- Preservation of universal sentences under substructures in higher-order logic December 5, 2025That "A universal sentence is preserved under substructures" seems to be a well-known fact in first-order model theory. I am dealing with a theory that quantifies over functions and relations, and I was wondering if it is true that the same result holds.user1868607
- Is Simple path relation $P_n$, length of simple path n between x and y, definable by $P_3$ and $P_4$ in first order logic, with no edge relation [closed] December 5, 2025Consider FO over graphs, but instead of the usual binary relation symbol $ E $ interpreted as the edge relation, there are two binary relation symbols $ P_3, P_4 $. The symbol $ P_3 $ (resp. $P_4 $) is interpreted as the set of pairs $(x, y)$ such that there is a simple path of […]hulululu_Kaveman
- Can Direct Proof be used in P: false => Q: false? [closed] December 3, 2025I'm reading "Basic Proof Techniques" by David Ferry (https://www.cse.wustl.edu/~cytron/cse547/IntroToProofs_Final.pdf) I don't know how authoritative it is, but the document says The steps of Direct Proof (Proof by Construction) are Assume that P is true. Use P to show that Q must be true. If the document is correct, can I apply the steps there to […]Jason Cho
- Are propositions involving undefined expressions (like $x/(x-1)=0$ at $x=1$) always assigned a truth value in first-order logic? [duplicate] December 3, 2025In ordinary mathematics, an expression like x/(x-1) = 0 is considered undefined at x = 1. In classical first-order logic (for example, as presented in Mendelson), can we legitimately say that the formula is false at x = 1, or must we treat it as neither true nor false because the term x/(x-1) is undefined […]Toxic Pusher123
- does second-order elementary equivalence coincide with isomorphism for two structures December 3, 2025Let $A$ and $B$ be two structures. We say they are second-order elementarily equivalent ($A \equiv_2 B$) if they satisfy exactly the same second-order sentences. Is this true or not? $$ A \equiv_2 B \;\Longleftrightarrow\; A \cong B. $$ If it is true, Why is this the case? I understand that in first-order logic, elementary […]Mr. ruler
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