Questions tagged [first-order-logic]
For questions about formal deduction of first-order logic formula or metamathematical properties of first-order logic.
9,967 questions
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Does the Axiom Schema of Separation in any sense "provide" only countably many definitions of subsets of $\mathbb{N}$?
Section 6 (pg. 24-25) of this paper contains the following excerpt (bold for emphasis added by me):
The axiom of Separation could also be called the axiom of Definable Subsets. A subset $T$ of $S$ is ...
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How do I turn the sentence into predicate calculus? [duplicate]
How do I write the sentence " Ana knows every one of Bob's friends" in predicate calculus?
I understand "knowing a person" and "being friends with person" can be two ...
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Relation between Tarski's conception of truth and (implicit) Axioms
I am trying to understand the original paper of Tarski Concept of Truth in the formalized languages, as printed in his collected works.
I have read introductory texts from Shoenfield Mathematical ...
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Under what conditions does a proper class model establish relative consistency?
Say $\mathcal L$ is a lexicon of first-order logic with equality. $\mathcal L$ is finite and may contain constants, functions or relations.
$C(x)$ is a predicate of the language obtained from $\...
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Uniqueness quantification example
I have encountered a quite simple sentence, which is:
The polynomial $x^2-6x+9$ has exactly one root in $\mathbb{R}$.
If I were to write this using quantifiers it would obviously be:
$\exists!_{x \in \...
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Semantically "undefined" terms of non-standard length in FOL with $\omega$-nonstandard metatheory
As preliminary background, the following passage from Foundations of Set Theory by Fraenkel et. al.:
The object-language of a given theory is sometimes an artificial symbolic language...When an ...
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Indiscernible sequence which cannot be extended
Take $\alpha=\omega +1$,$\Delta=\{\phi(x,y)\}$ and let $\mathfrak{C}\models\phi[a^i,a^j]$ iff $i<j<\omega+1,\mathfrak{
C}
\models(\neg\exists y)\phi(a^\omega,y);$ then $\{a^i:i<\omega+1\}$ is ...
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Can first-order logic prove "theorem schemas"?
Examples of "axiom schemas" are well known, e.g. Replacement in ZFC. The notion I have of "theorem schema" is an analogous notion, i.e. an infinite collection of (non-axiom) ...
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Is the formula $\forall xR(x,y)\rightarrow\exists x\neg R(x,y)$ from "A First Course in Logic" truly unsatisfiable? (erratum)
I am reading A First Course in Logic by Shawn Hedman and came across a statement on page 64 that I believe is incorrect. The book presents the following formula as an example of an unsatisfiable ...
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Justifying universal generalization after existential elimination using dependence between quantifiers
In an attempt to transform following equivalent definitions of the cartesian product from the first into the second, trying to be rather formal about it:
$$
\forall A : A \in X \times Y \...
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What is the exact definition of term in first-order logic?
Suppose t is a term, $f_n, n\in\mathbb{N}$ are function symbols, I have no doubt that $f_1(t),f_1 f_2(t),f_1 f_2 f_3(t)$ are all terms. But what about these:
$f_1 f_2 f_3...f_n(t)$ (note that n is ...
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Generic way to combine formal systems (e.g., first-order theories)?
Lately I’ve been wondering if there was a generic way to combine two formal systems.
That means, we have formal systems 1 and 2 which talk about objects of type 1 and 2 respectively, and we want to ...
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Prove $\forall y \exists x (Fx \leftrightarrow \neg Fy )\rightarrow (\exists x Fx \land \exists x \neg Fx)$ in Open Logic Project system
How to prove the following $\forall y \exists x (Fx \leftrightarrow \neg Fy )\rightarrow (\exists x Fx \land \exists x \neg Fx)$
I am kidn of stuck I am using the open logic calculator
https://...
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Prob. 17 (c), Sec. 1.5, in Rosen's DISCRETE MATHS, 8th ed: How to translate this statement into a logical expression [duplicate]
From Prob. 17 (c), Sec. 1.5, in the book Discrete Mathematics and Its Applications by Kenneth H. Rosen, 8th edition:
Express the following system specification using predicates, quantifiers, and ...
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How to formalise “There are two students who between them have chatted with everyone else”?
From Discrete Mathematics and Its Applications by Kenneth H. Rosen, 8th edition :
Let $C(x, y)$ be the predicate “$x$ has chatted with $y$”, where the domain for $x$ and $y$ consists of all the ...
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Is the first-order theory of the average binary operation on the reals finitely axiomatizable?
Looking at the question What is a finite basis for the identities of the average binary operation on the reals? I found myself wondering: if we extend to the first-order theory of the average ...
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Translating "Some triangles are green" [duplicate]
I have a pretty basic but tricky question about predicate logic.
$T(x) : x$ is a triangle
$G(x) : x$ is green
Using the above predicates, I am pretty sure that the translation of the sentence "...
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How to properly use substitution in first-order logic? [duplicate]
Let's say I want to substitute $x$ for $y-1$ in statement $(\forall x)(x>0)$. So I should get $(\forall y)(y>1)$.
What is the logical basis for getting the result?
It seems that I need to prove ...
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Can ZF be axiomatised using first-order logic using a finite number of axioms?
ZF is known to contain a number of axiom schemas which make the axiomatisation not finite. However, the following paper of Kleene
Stephen Cole Kleene: Finite axiomatizability of theories in the ...
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Proving that Rosser's sentence and Gödel's sentence are not provably equivalent in Peano Arithmetic
I am doing exercise 3 in section 6.6.1 in "A Friendly Introduction To Mathematical Logic" (see the original description here). I am stuck on the following exercise. Any hints or help would ...
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How to symbolise "if x is a real number, then $⌈x⌉−⌊x⌋= 1$ if $x$ is not an integer and $⌈x⌉−⌊x⌋= 0$ if $x$ is an integer"?
How to symbolise "if $x$ is a real number, then $⌈x⌉−⌊x⌋= 1$ if $x$ is not an integer and $⌈x⌉−⌊x⌋= 0$ if $x$ is an integer"?
Predicates:
$P(x) :$ x is a real number
$Q(x) : ⌈x⌉−⌊x⌋= 1$
$R(x)...
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Proving that recursivly defined functions defined using $\Delta$-formulas are representable
I am doing exercise 2 in section 5.10.1 of "A Friendly Introduction to Mathematical Logic", (exercise can be found here), and i cannot find a solution. I have tried making my own solution, ...
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Countability of models of ZFC
I'm reading about Skolem's paradox and the idea that countability in set theory is "relative to a model", and that makes we wonder what is actually meant by saying that a model of ZFC is ...
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Speaking constructibility of a natural number in ZFC
This question is related to this earlier question Precise meaning of a natural number being independent of ZFC.
Now my intension is to formalize the sentence “we can construct a natural number $n$ ...
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Reducing number of axioms used
Consider ZF set theory, with all the usual alphabet and logical symbols.
Let $\psi$ denote the axiom of regularity:
$$\psi\equiv\forall x(x\neq\emptyset\rightarrow(\exists y\in x)(y\cap x=\emptyset))$$...
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