Tag: OpenLab 8
Logic on Math StackExchange
- Is non-Noetherianness first order axiomatizable in the language of commutative unital rings? March 30, 2023By a standard argument* using Łoś's ultraproduct theorem (either directly or through the compactness theorem), Noetherianness is not first order axiomatizable in the language of commutative unital rings. Is the same true of non-Noetherianness? (Of course, it must at the very least not be finitely axiomatizable.) I've tried the obvious thing but have gotten nowhere. […]Rafi
- What ZF can do and Peano's axiom cannot. March 30, 2023I am interested in how much math can be done from Peano's axioms and what can't. What is there in the mathematics done with ZF that cannot be done with Peano's axioms?Hayatsu
- Specifying Calculus of Constructions (or something similar) in LF March 29, 2023Consider LF, the logical framework used to define UTT (unified theory of dependent types). The next two quotes are from here. LF is a simple type system with terms of the following forms: $$\textbf {Type}, El(A), (x:K)K', [x:K]k', f(k),$$ where the free occurences of variable $x$ in $K'$ and $k'$ are bound by the binding […]user125234
- Are “if / iff” interchangeable for “when / whenever”? March 29, 2023Let $A$ and $B$ be arbitrary sets and consider the following two statements: \begin{gather} (x\in A)\Rightarrow (x\in B)\\ (x\in A)\Leftrightarrow (x\in B) \end{gather} These two statements are usually worded as follows: If $x$ belongs to $A$, then $x$ belongs to $B$” “$x$ belongs to $A$ if and only if $x$ belongs to $B$”. My questions […]Hector
- How do Robinson arithmetics axioms prevent this model of N? March 29, 2023In the Peano arithmetic wikipedia article, a figure is shown on why the axiom of induction is necessary, without it, the set of whole dominos would be a valid representation of N In Robinson arithmetic, we have the axiom y=0 ∨ ∃x (Sx = y) instead. I don't see how this axiom prevent that situation, […]user2370139
- Changing EM blueprints in Abstract Elementary Classes March 29, 2023I'm currently reading some notes on AECs (here), and just wanted to make sure I understand the argument in Corollary 12.8, for I'm still getting used to generalized Ehrenfeucht-Mostowski models in AECs. As far as I understand, EM blueprints/templates $\Phi$ keep track of the quantifier-free sentences some indiscernible satisfies (that is, a single $n$-type for […]interregno
- In the class, there are $30$ students March 29, 2023I am preparing to take an entrance exam for a university in my country, which will happen soon. As part of my preparation, I have been practicing with some sample math tests provided by the university. However, I recently came across a math problem that I could not understand at all. This is the exercise: […]Simebe
- Why do we care about the decidability of a theory? March 29, 2023I'm new to logic theory and i'm stuck with the understanding of decidability of a logical system. As far as I understand, the decidability of a logical system is related to the existence of an effective (in terms of time) procedure to tell whether a given formula can be proved by using the "instruments" of […]InTheSearchForKnowledge
- Free variables in derivation systems March 29, 2023I work relative to a fixed first-order language $L$. As usual I call those formulas which do not contain free variables sentences. There are several derivation systems for first-order logic. For example, in Goldrei's Propositional and Predicate Calculus a derivation of $\psi$ relative to a set $Ax$ of sentences is a list of formulas $\phi_1,....,\phi_n$ […]Nico
- Is every limit computable set in the Ershov hierarchy? March 29, 2023A set is limit computable, or $\Delta^0_2$, if its characteristic function is equal to $\lim_{s\rightarrow\infty}g(x,s)$ for some total computable function $g$. And given a computable ordinal $\alpha$ and a path $P$ through Kleene's $O$, a set is $\alpha$-c.e. if its characteristic function is equal to $\lim_{s\rightarrow\infty}g(x,s)$ for some total computable function $g$ such that there […]Keshav Srinivasan
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