Week 3Â Assignments
Written work â Sec 1.8 p.29: 3, 5, 6, 8, due Tuesday, 9/17 (in class)
WeBWorKÂ – Start on WeBWorK 3, (it will not be due until Tuesday, 9/24 at midnight).
OpenLab – OpenLab #3, due Thursday, 9/19 (at start of class).
Logic on Math StackExchange
- Preservation of Universal Sentences Under Finite Substructures in Higher Order Logic April 19, 2026First off, I apologize if my question is considered elementary or perhaps even phrased incorrectly. I wanted to ask if "universal sentences that quantify over functions and relations preserve under substructures" holds true for finite models. I did come across this question but I was not sure if this held for finite models. The reason […]user1438127
- .Formalizing the Provability of Unprovability in Lean 4: A Derivation via Löbâs Theorem [closed] April 16, 2026I am investigating the metalogical constraints on a consistent, recursively axiomatized theory T (extending Peano Arithmetic) regarding its ability to prove sentences about its own unprovability. Specifically, I've been looking at whether T can prove ÂŹProvT(âSâ) for some sentence S. and how this relates to the Second Incompleteness Theorem. To explore this rigorously, I formalized […]Sanand
- Can Infinite Time Turing Machines simulate computations of Ordinal Turing Machines for a specified countable time? April 16, 2026From the replies to one of my questions on Mathoverflow, I can conclude that Infinite Time Turing Machines can somehow simulate computations of Ordinal Turing Machines, but the whole situation is not clear to me. In this question, the term ârealâ implies an infinite binary sequence of length $\omega$. We assume that there are: (i) […]lyrically wicked
- Terminology question: why "positive" in "contrapositive"? April 13, 2026The contrapositive of a statement "if $P$ then $Q$" is "if not $Q$ then not $P$". In Chinese "contrapositive" is translated to "conversion-negation" because it first converse the statement (if $Q$ then $P$) and then negates $P$ and $Q$. This makes me confused why in English it is called "contrapositive". Does "contra" correspond to "conversion"? […]Asigan
- Do we say a (deductive) argument is satisfiable? April 12, 2026A formula is satisfiable if it is true under some assignment of values to its variables. (source) A formula is valid if and only if it is true under every possible interpretation. (source) In terms of validity, deductive arguments may be either valid or invalid. An argument is valid, if and only if (iff) it […]Jason Cho
- List of statements in logic [duplicate] April 10, 2026In a book of basic propositional logic I do not expect any naturals. Nevertheless I see quite often "listsâ of statements like $q_1,\cdots,q_n$. Neither âlistâ nor $n$ are defined. The existence of the naturals is derived from set theory and logic. How can these "listsâ of statements be defined in logic without using the naturals?BW M
- Set theoretic definition of pre-image April 9, 2026Let $f : X \to Y$ and let $B \subseteq Y$. Then according to Wikipedia, the preimage of $B$ under $f$ is defined like \begin{align*} f^{-1}\bigl[B\bigr] = \bigl\{ x \in X \;:\, f(x) \in B \bigr\} \,. \end{align*} By definition, we have: \begin{align*} x \in f^{-1}\bigl[B\bigr] \iff x \in X \;\land\; f(x) \in B \,. […]Jacob Lockard
- Gödel's incompleteness theorems as a functional program April 8, 2026In the talk "Propositions as Types" given by Phillip Wadler, Wadler gives an overview of Gödel's incompleteness theorems, and then at 3:00, says: "Don't worry about the details about how he did [the theorem], although it is one of the world's first functional programs..." I was not aware of a connection between the incompleteness theorems […]tlonuqbar
- Is a model just a set of symbols in a meta-language? April 7, 2026Lately, Iâve been reviewing some basic concepts of logic. And a question came to mind regarding the models of a theory. My questions: As I understand it, in order to talk about models, we also need a meta-language or meta-theory. Otherwise, how one can define what a model is without invoking an âexternalâ (a stronger) […]3m0o
- Non-trivial examples of inconsistent theories April 6, 2026Is there any example of an inconsistent axiom system where the inconsistency is not trivial? Iâd like an example where you could think at first that this theory is not a trivial one or even that it is useful. To be more concrete, I want to distinguish trivial and obvious. By trivial, I mean technically […]QuestioningEverything
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