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- 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
- Nuances of logical connectives in defining an operation July 27, 2026From a German textbook on set theory: Let $F$ and $G$ be unary operations. Show that the mapping $F \circ G$ is an operation. Use any axiom system you like that allows you to define $F$ and $G$ as operations. Do the same for the mapping $H$ defined as: $$H(x) := \begin{cases} F(x), & \text{if […]Max
- How should the phrase “arbitrary but fixed” be understood? [duplicate] July 26, 2026For the proof of the following proposition: $$ x_n>0,\qquad \lim_{n\to\infty}x_n=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\ln x_n=-\infty, $$ there are actually two limiting processes. Taking the solution shown in the image as an example, here $G$ is a number greater than $0$ that is said to be “arbitrary but fixed.” Then is $G varying and arbitrary? (I am sorry, […]Michael James
- Intended Interpretation: Fact or Fantasy? July 23, 2026Regarding the so-called Intended Interpretation (a.k.a. standard model) of Peano Arithmetic (PA), the term is occasionally used not in a relative sense (relative to another, nonstandard, model) but rather in an absolute sense. Let's denote such an entity X. What exactly is meant by this? More specifically, does X satisfy all the axioms of PA? […]Mikhail Katz
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