Tag: calculus
Logic on Math StackExchange
- Is learning STT before DTT a good idea? August 12, 2026I have a good knowledge of category theory and have become interested in its connections to logic. My goals are to understand how the (fibred) categorical structure (e.g. $\prod, \Rightarrow$) concretely corresponds to the logical ($\forall, \to$) and type-theoretical ($\Pi$ (only possible in DTT, I think), $\to$) structures. In particular, I would like to understand […]QuestioningEverything
- On the undecidability of an iterative parity-twisted divisor-mapping sequence August 12, 2026I have constructed an arithmetic function that exhibits a chaotic behavior reminiscent of Collatz-like dynamical systems, but with a feedback loop driven by the partitions of parity-shifted divisor geometries. The system appears to inherently embed the Halting Problem within standard multiplicative number theory. I am seeking insights into whether this structure is provably independent of […]Knut Sylvén
- Question regarding L-implication and L-truth August 12, 2026Referring to Theorem T6-1a and Theorem T6-1b, from Carnap, are they just using different terminology to assert the same notion, or are they in fact saying different things? My understanding of Theorem T6-1a: If a sentential formula G is L-implied by an L-true sentential formula H, then G is true under every interpretation for which […]Los Angeles
- Does Cantor's diagonal condition inherently demand $x \notin L$ to avoid being an ill-formed execution? [closed] August 6, 2026To see the structural nature of the diagonal condition, let's test it on a simple finite list of $3$ elements: $L = \{L_1, L_2, L_3\}$. The diagonal condition constructs a number $x$ such that: For all $n \in \{1, 2, 3\}$, the $n$-th digit of $x \neq$ the $n$-th digit of $L_n$. Let's test if […]Lumen Croft
- Can there exist a universal set containing all legitimate sets. August 5, 2026I've updated this question to reflect my current understanding after reading the answer given by Michael. The main doubt is why the proposed "filter first, then collect" approach fails and how Gödel's and Turing's results relate to it. I understand Russell's paradox and why the collection of all sets that do not contain themselves cannot […]Zoya Ali
- Is Wikipedia's $\epsilon$ - $\delta$ limit definition standard? August 2, 2026So often I see that when the $\epsilon$ - $\delta$ definition of a limit is presented, they forget to quantify the $x$-variable in their definition (source: Thomas' Calculus: Early Transcendentals, 15th Edition, by Joel Hass, Christopher Heil, Przemyslaw Bogacki, and Maurice Weir, pg. 75): $\style{font-size: px;}{\color{#0090D3}{\boldsymbol{\mathsf{DEFINITION}}}}$ Let $f(x)$ be defined on an open interval about […]Bob Marley
- The fear of paradoxes [closed] August 1, 2026In the turbulence of the paradox of Russell a lot of ideas came up, as type theory, combinators and lambda calculus, which have been successful in computer science. And computers eventually became a sort of method of mathematical proof processing (Lean etc). Despite this, or perhaps even more because of it, I am surprised by […]Lehs
- A question about Kunen's definition of $\leq$ and the proof of Lemma II.1.3(4) August 1, 2026I am reading Kenneth Kunen's Set Theory (2011 edition), Chapter II. Kunen defines $$ \Gamma\lhd\Lambda \quad\Longleftrightarrow\quad \Lambda\vdash\operatorname{Con}(\Gamma), $$ and defines $$ \Gamma\leq\Lambda $$ to mean that there is a finitistic proof of $$ \operatorname{Con}(\Lambda)\rightarrow\operatorname{Con}(\Gamma). $$ He later remarks that “finitistic” may be understood as “formalizable in Primitive Recursive Arithmetic (PRA).” I found the following related […]mathlogic2025
- Why is the statement "for all real numbers $s$, there exists a real number $t$ such that $t > s$" true? July 29, 2026This statement showed up in my text alongside the statement, "there exists a real number $t$, such that for all real number $s$, we have $t > s$." The first statement is true while the second statement is false. The text argues that it is because of the order in which the statements appear, such […]Mezzoforte
- “Equivalent statements have the same truth tables”? July 29, 2026My textbook states, “$ P{\iff} Q$ is true precisely when $P$ and $Q$ are equivalent statements. Hence, equivalent statements have the same truth tables.” This statement is confusing: how can $P$ and $Q$ have the “same truth tables”? I envision the truth tables as: P Q Logical statement T T ? F F ? I […]Mezzoforte
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