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Logic on Math StackExchange
- Does this article misapply a result in mathematical logic? March 13, 2026A recent neuroscience paper (Milinkovic & Aru 2026, Neuroscience and Biobehavioral Reviews) argues that biological computation differs fundamentally from digital computation, partly on the basis of Tarski's 1951 result on real closed fields. One of the representative passage reads: "Whilst discrete natural number arithmetic (the foundation of digital computers) is fundamentally marked by incompleteness and […]Daniel Lü
- What class constructions can we do in Ackermann set theory? March 11, 2026Consider Ackermann set theory: Everything is a class, and some classes are sets. A1 (Extensionality). Classes are equal iff they have the same elements. A2 (Comprehension). For any predicate $\phi (x)$ (possibly with parameters), there is a class whose elements are all sets $x$ such that $\phi (x)$ holds. A3 (Heredity). Any element or subclass […]Zhen Lin
- Discrete math. Necessity and sufficiency [duplicate] March 11, 2026Let p: Climbing is safe q: Weather temperature is less than 0 r: The guide is familiar with the mountain. Express the following statement using symbols “For climbing to be safe it is necessary but not sufficient that the weather temperature is above 0 and your guide is familiar with the mountain.” This question popped […]Kareem Qablawi Kareem
- Propositional logic: $(S \vdash R) \vdash [(S \vdash \neg R) \vdash \neg S]$ [closed] March 11, 2026This was a question on an assignment I have for my intro to logic class, it has no premises, how do I even approach solving this?newguy8
- Can we force $0^\sharp$ to exist? March 8, 2026So Cohen forcing adds real numbers, and the set of indiscernibles for Gödel's universe $0^\sharp$ can be coded as a real number (the code of all formulas satisfied by the first $\omega_1$ elements of the indiscernible sequence, hence a subset of $\omega$). Can we use Cohen forcing, a variation, or any other forcing to force […]edgar alonso
- Show that the well-ordering theorem implies the axiom of choice - are equivalence classes permitted March 8, 2026After a bit of searching around, I have not seen this kind of proof (if it is in fact one) which shows that the well-ordering theorem implies the axiom of choice, so I thought it best to have it checked by others. Proof : Let $A$ be an arbitrary set. Let $\sim$ be an equivalence […]Tarran M
- Intuitionistic logic as the 'minimum' logic necessary for mathematics - and how it relates to topoi's significance March 5, 2026It seems like we can do a great deal of mathematics constructively, i.e. using intuitionistic logic. Where intuitionistic logic is insufficient, we use a strengthened form (classical logic), for example to construct non-continuous functions $\mathbb{R}\to\mathbb{R}$. Many mathematicians uncritically/implicitly use classical logic to do all of their work, and the ones who examine their language critically […]Amitai
- Is a completely consistent foundation of mathematics possible? March 4, 2026According to the incompleteness theorems, any sufficiently expressive axiomatic system cannot prove its own consistency, this leads to an infinite regression problem, for instance to prove Con(PA) we use ZFC, but this proof cannot be taken seriously unless one proves Con(ZFC) and this continues indefinitely. Is there research in the foundation of maths which attempt […]Davyz2
- Prove $\forall x(A(x)\rightarrow A(x))$ in minimal logic without using the Deduction Theorem March 4, 2026I am reading An Introduction to Proof Theory by Mancosu et al. I tried one of the exercises (p. 52), but I got stuck: Prove without using the deduction theorem $$\vdash_{M_1}\forall x(A(x)\rightarrow A(x))$$ (the book's '$\supset$' is replaced with the now-standard '$\rightarrow$' henceforth; the subscripts ${M_0}$ and $M_{1}$ denote minimal propositional and predicate logic, respectively) […]Theodore Bucklseworth
- When we give a definition in math and say "otherwise, ..." does the definition precedes the "otherwise" case? This book seems to apply that rule March 3, 2026When we define an objet, say $I^=$ (a subset of indices), with a definition. And if we say: otherwise, the index, if not in $I^=$, is in $I^niobium
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