If I was invited to give advice to entering students on the first day of Mat 2071 I would tell the students to always be prepare for class, and pay attention to the teacher when he or she is explaining the course. I would tell them to come in class at time, and do not miss a lot of days in order for them to keep in track. I would also advise them to always ask questions when they don’t understand something, and keep practicing the subject in class and also at home which would help them to progress and also succeed in the subject.
Logic on Math StackExchange
- Are arguments of the form $A \rightarrow \neg A$ used in proofs by contradiction? June 14, 2026In proof by contradiction, the standard pattern is: Assume $A$ Derive a contradiction (e.g. $B \land \neg B$ or $0 = 1$) Conclude $\neg A$ My question concerns arguments that appear to establish: $$ A \rightarrow \neg A $$ Is this ever the actual structure of a proof by contradiction in mathematics? More specifically: If […]Antonios Sarikas
- What conclusion can be derive from this statement [closed] June 14, 2026I was answering some questions and I stumble upon this: "Some graduate are coders. Some coders are data scientist. All data scientist are ML Experts logic question" I am being given with 5 options: A. Some ML experts are graduate B. Some data scientist are graduate C. Some ML experts are coders D. All of […]MisterMuffin
- Algebraic homogeneity in saturated models June 12, 2026Let $M$ be a countable saturated model in a countable language. Let $\bar{a}$, $\bar{b}$ be tuples in M. If $c \notin \mathrm{acl}(\bar{a})$, does there exist an automorphism $f$ that fixes $\mathrm{acl}(\bar{a})$ and $f(c) \in M \setminus \mathrm{acl}(\bar{b})$? In an $\omega$-categorical structure, the algebraic closure is finite, but in general, we cannot use homogeneity in a […]Focaccia
- Does $ZC$ proves $\mathrm{Con}(ZC_{\text{fin}})$? June 11, 2026It's a well known result that $ZFC\vdash\mathrm{Con}(ZFC_{\text{fin}})$ (where $ZFC_{\text{fin}}$ denotes $ZFC$ without the Axiom of Infinity), wich is usually proved by showing that $V_\omega\vDash ZFC_{\text{fin}}$. However, as this other answer shows, in order to construct $V_\omega$ we need The Axiom of Replacement. Naturally, this leads me to the following two questions: Even without Replacement, can […]Xennonio
- Is the club filter on $\omega_1$ normal under $\mathsf{AD}$? June 7, 2026I know that $\mathsf{ZF+AD}$ proves the club filter on $\omega_1$ is an ultrafilter, and therefore a normal ultrafilter if we also assume $\mathsf{DC}$, because the usual "catch your tail" argument shows that if $(A_\alpha:\alphan901
- Can these six equivalence laws derive all others, and are any of them redundant? June 3, 2026I'm trying to make a card game about second order arithmetic. As a proof of concept, I'm making a simplified prototype using only propositional logic statements and equivalence laws. I want a set of equivalence laws that are as simple as possible without missing out on any functionality. In a logic class I took a […]SpearmintHaiku
- How to choose between creating axioms vs. using conditional vs. simulation? June 2, 2026Axiom vs. Conditional From natural deduction, we can convert between $$A_1, \ldots, A_n \vdash C$$ and $$\vdash (A_1 \land \cdots \land A_n) \rightarrow C$$ So technically, we can have an axiomatic system with zero "non-inference-rule" axioms. Technically, entire mathematical theories can be built with theorems preceded by super-long suppositions. Is it a question of ontology? […]Ming Lin
- Characterisation of ordered pairs May 30, 2026What follows is a complete rephrasing of the original question, mostly taking @Noah Schweber's answer into account. Starting out with FOL and the axioms of $\mathsf{ZF}$ set theory, one is able to define ordered pairs in quite a few different ways. Regardless of any individual definitions, there is the general property that we require of […]ZeroAlephZeta
- Why do we need to re-prove theorems that have been proved already? Why can't theorems be taken as axioms? May 28, 2026Special thanks to MikhailKatz for suggesting the right questions to me! I have several questions about theorems: Why do we need to re-prove theorems that have been proved already? Why are students expected to learn proofs of theorems that have been proved long ago (and can therefore be taken as axioms)? Thanks.Mike_bb
- If there is a proof of $0 = 1$ in Peano Arithmetic, can we say anything about its length? May 21, 2026Let $M$ be a model of $\text{PA} + \neg\text{Con}(\text{PA})$, and let $c \in \mathbb{Z}$. Define $i(M)$ as the minimum number of symbols in any proof of $0 = 1$ (note, this "proof" will have nonstandard length), and let $f(M, c)$ be the set of standard primes dividing $\lfloor \log(i(M)) \rfloor + c$. Is every set […]Beren Gunsolus
Hi Miralia,
Thanks for the response – that’s great. I’d prefer that you leave it as a comment on the OpenLab #1 assignment, here:
https://openlab.citytech.cuny.edu/2017-fall-mat-2071-reitz/?p=292
(scroll to the bottom and you’ll see that you can leave a reply – just copy/paste it in there). This will ensure that it gets counted correctly for your grade.
Best,
Prof. Reitz