Tag: Wau
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- When are parentheses actually needed in formulae in first-order logic? August 14, 2026Suppose $\mathcal L$ is a vocabulary consisting of function symbols and relation symbols of given arities (we regard constant symbols as function symbols of arity $0$). Recall that the set of $\mathcal L$-terms is the smallest set $T$ such that: Every variable is an element of $T$, If $f$ is an $n$-ary function symbol, and […]Joe
- A categorical description of local finiteness August 14, 2026A variety (in the sense of universal algebra) is called locally finite if every finitely generated free algebra is finite. Recently I have been interested in those properties of varieties which are preserved by categorical equivalences, and I realised that the property of being locally finite is such. This can be deduced from the following […]Carlyle
- 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
- 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
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