Written work – None.
WeBWorK – Assignment #5, due Tuesday, October 9th, at midnight.
OpenLab – OpenLab #4, due Thursday, October 11th, before class.
Logic on Math StackExchange
- Are the Turing degrees on Cantor exactly the same on Baire? October 15, 2024Background: As far as I know, the Turing degrees are usually defined (considering sets of naturals) on the Cantor space $2^\omega$ as the equivalence classes induced by the Turing reducibility $\le_T$. That is for $X \in 2^\omega$: $$ \deg_T^{2^\omega}(X) := \{ Y \in 2^\omega \mid Y \equiv_T X \} = \{ Y \in 2^\omega \mid […]NON
- Looking for a textbook on logic October 14, 2024I'm preparing an introduction to logic for non-mathematicians, and in the process, I’m trying to better understand the basics myself. I came across a post on this forum: Difference between $\to$, $\models$ and $\implies$, where a textbook is mentioned but not directly referenced. In the post, @MauroALLEGRANZA provided an answer, also citing the textbook, but […]mandel_broetchen
- Does proof by contradiction construct a vacuously true statement? October 14, 2024When you are proving something using proof by contradiction, are you actually constructing a vacuously true statement? Let's say that you want to prove $P$ via proof by contradiction. First, you assume that $\lnot P$ is true. Then via direct proof, you show that $\lnot P \Longrightarrow \bot$. Since the only way for the statement […]Jon
- In the positive fragment of Naive Set Theory, does $\{x:x\in x\}$ belong to itself? [duplicate] October 14, 2024The Naive Set Theory ($\mathsf{NST}$) is an inconsistent, trivial theory, but its positive fragment was shown to be consistent and is basis of e.g. positive set theory $\mathsf{GPK}_{\infty}^{+}$ of Olivier Esser. In the positive fragment, instances of unrestricted comprehension are restricted to only positive formulas (the smallest class of formulas containing atomic membership and equality […]Timotej Šujan
- Confusion with the proof for the Halting Problem October 14, 2024I was considering the halting problem, and I'm confused about why it is impossible. As I understand it, the problem is defined as follows. There exists a program $h(x,i)$ which takes as input a program $x$ and an input to that program $i$ and then reports if the program $x$ reaches a final decision on […]Emizaquel
- What if someone proves Self referencal paradox in general [closed] October 13, 2024Good greetings I have a question that might seems stupid to one yet Is it possible to find a general solution for self referencal paradox??? Like for say a general algorithm that can help to find out the way to solve any/almost all self referencal paradox I would be incredibly grateful if one could provide […]Avantika Vats
- Is it true that there are real numbers that cannot be expressed? [duplicate] October 13, 2024This might be elementary (or obviously wrong) for mathematicians. I am an engineer. Since we write numbers using finite strings of symbols (not necessarily digits - even formulas are finite strings of symbols) does that mean that there are a lot more real numbers which cannot be written in any way, than there are real […]Censored to protect the guilty
- What exactly is (should be) a “structure” for $L_2$ (language of 2nd-order arithmetic)? October 13, 2024From Simpson’s Subsystems of Second Order Arithmetic: The language of second order arithmetic is a two-sorted language…Variables of the first sort are known as number variables…Variables of the second sort are known as set variables…Atomic formulas are $t_1 = t_2$, $t_1 < t_2$, and $t_1 \in X$ where $t_1$ and $t_2$ are numerical terms and […]NikS
- Prove: Neighbourhood of Spanning Tree is exact October 12, 2024In my lecture the neighbourhood of a spanning tree f is defined as: N(f) = {g element of F | g can be obtained from f as follows: Add an edge to the tree f, producing a cycle, then delete any edge from the cycle}. Where F is the set of feasible solutions. In the […]Franka
- $∀xP(x)∧Z(x) \equiv ∀x(P(x)∧Z(x))\,?$ [closed] October 11, 2024Without more parentheses, I don't know the scope of the universal quantifier in $$∀x P(x)∧Z(x).$$ Does it apply to both $P$ and $Z:$ $∀x(P(x)∧Z(x))$ ?Bob School
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