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Logic on Math StackExchange
- 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
- What is Enderton's definition of tautological implication in first-order logic? March 2, 2026Theorem 24B of Enderton's Introduction to Mathematical Logic states: $ \Gamma \vdash \varphi \text{ iff } \Gamma \cup \Lambda$ tautologically implies $\varphi$. But as far as I can tell he never defines tautological implication for first-order logic. He does offer a definition within the sentential logic framework. My assumption is that we have tautological implication […]theQman
- Hi im 14 years old learning in my own [closed] March 2, 2026I am a 14-year-old student exploring formal systems and set theory on my own. Below are my notes based on my current understanding. I would really appreciate precise feedback, especially if there are conceptual mistakes or ambiguities. The following is my notes. (I translated my notes from Indonesian to English): An axiom is a basic […]do gaming
- Splitting a manipulated equation into two independent equations March 1, 2026Let us restrict attention to equations in one real variable $x$ satisfying: The solution set is nonempty (i.e., not of the form $1+x=x$). Consider the equation $$ (x-1)^2 = 0, \tag{1} $$ which has solution set ${x = 1}$. This is equivalent to $$ x^2 - 2x + 1 = 0. \tag{2} $$ Now suppose […]Neumann
- Reference request: A formulation of Gödel's system $T$ without formulas March 1, 2026Page 2 of Dowek's "Gödel's system $T$ as a precursor of modern type theory" gives a presentation of system $T$ as a set of terms with a rewrite relation: The modern definition of the system $T$, as a set of terms, has been given later by Tait: the fact that Gödel system $T$ can express […]C7X
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