2
votes
0answers
42 views

Path-independent contour integrals and how to define them

If a contour $C$ is parameterized by $z(t): [a, b] \to \mathbb{C}$, then we define $$ \int_C f(z) \, dz= \int_a^b f(z(t)) \, z'(t) \, dt.$$ If the contour integral on the left side is equal to some ...
1
vote
3answers
98 views

Why do we formalize conceptions?

Why do we always try to formalize conceptions? Let's take the naive conception of sets, why do we try to write down a list of axioms? what do we earn in doing so? I'm looking especially for ...
3
votes
1answer
79 views

Semantics and Logical structure in Definitons

Continuation of Free and bound variables in "if" statements definitions: A number is even if it is divisible by $2$. The number is even if it is divisible by $2$. Is the usage of the ...
1
vote
1answer
44 views

Logical structure of definitions

Here are some "concepts" that are confusing me: The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion ...
1
vote
2answers
53 views

The Definition of the Indicative Conditional

From Wikipedia, we have: In natural languages, an indicative conditional is the logical operation given by statements of the form "If A then B". Unlike the material conditional, an indicative ...
2
votes
2answers
77 views

In Mathematical Logic, What is a Language?

I've been reading about mathematical logic and computability theory, but I'm somewhat confused on one note: what exactly is a language? What does it mean when I am told "let $\langle 0, +, \leq ...
3
votes
1answer
73 views

Definition of intersection of a family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A: (A\in F \implies x\in A)\bigr\} $$ If my understanding serves me correctly this notation means that all the $x$ ...
1
vote
1answer
25 views

Reusing Variables First Order Logics

Assume we have a parametrized FO formula of this form: $$\varphi(x_1,x_2, y_1, \dots ,y_m) := \xi(x_1,x_2) \land \psi(y_1,\dots,y_m)$$ We want to use as few additional bound (quantified by $\exists$ ...
5
votes
3answers
170 views

What precisely is a vacuous truth?

Is there a proper and precise definition that goes something like this? Definition. A statement $S$ is a vacuous truth if ... ...
1
vote
1answer
67 views

What is the definition of “Winning Strategy” in an Ehrenfeucht-Fraïssé game?

I've read many descriptions and applications of a Winning Strategy, as much as many for a Strategy tout court, but when a formal, algebraic definition is called upon, I've found close to no input. I ...
1
vote
2answers
40 views

Help with understanding the definition of operation

I'm having trouble understanding this excerpt from Wikipedia, which defines an operation: Mainly, I don't understand what is meant by $V \subset X_1 \times...\times X_k$. Why does an operation ...
0
votes
2answers
71 views

I Need Help Understanding the Formal Definition of A Limit

I had been taught the formal definition of a limit with quantifiers. For me, it is very hard to follow and I understand very little of it. I was told that: $$\text{If} \ \lim\limits_{x\to a}f(x)=L, \ ...
1
vote
3answers
120 views

What is a formal definition of “predicate logic”?

I'm currently trying to get clear about some terms that are often used in computer science (I'm a computer science student), but were never formally introduced. Especially, I would like to know what a ...
1
vote
2answers
69 views

Meaning of variables and applications in lambda calculus

The wikipedia definition of lambda terms is: The following three rules give an inductive definition that can be applied to build all syntactically valid lambda terms: a variable, $x$, is ...
4
votes
1answer
40 views

Transforming Nested Fixed-Point Formulas into Infinitary Logic Formulas with Finitely many Variables

There is a definition (actually a description of how it could be defined) of a fixed-point logic formula. The formula is in inflationary fixed point logic (IFP) in this case but it could also be ...
0
votes
2answers
47 views

Is this predicate expressing compactness?

I don't quite know why definitions of "compact" use the expression "arbitrary collection". Am I correct in thinking that the following predicate is a definition of compact? Let $X$ by a set with ...
2
votes
1answer
157 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
2
votes
1answer
30 views

Generated equivalence relations in logics

Let $L$ be some logic (FO or stronger which is not important for this purpose). Given a $\tau$-structure $A$ and a formula $\varphi(x_1, \dots x_n) \in L[\tau]$ with free variables $x_1, \dots, x_n$. ...
0
votes
1answer
119 views

Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
4
votes
1answer
91 views

Why doesn't the definition of (model-theoretic) conservative extension need strengthening?

In the wikipedia page dedicated to conservative extensions, we find the following sentence: $T_2$ is a model-theoretic conservative extension of $T_1$ if every model of $T_1$ can be expanded to ...
2
votes
1answer
52 views

Is Wikipedia's definition of $\omega$-inconsistency problematic in this way?

I could be wrong, but the definition of $\omega$-inconsistency given at over Wikipedia seems slightly problematic. In particular, Wikipedia claims that $\omega$-inconsistency is a property of a theory ...
2
votes
2answers
213 views

What are authoritative publications regarding foundational mathematics?

I have a computer science background. In our world, there usually is an organization publishing standard documents for certain areas (e.g. W3C has Web standards, IETF publishes Internet-related ...
2
votes
2answers
99 views

Ted Sider's Definition of a Total Relation over a Set D

I'm working through Ted Sider's book "Logic for Philosophy," and I'm noticing some discrepancies between the definition of a "Total Relation" that he uses and the definition used in other places, ...
3
votes
3answers
190 views

what are first and second order logics? [duplicate]

The only knowledge I have on logic is due to a book I read a couple of years ago called Introduction to logic: and to the methodology of deductive sciences by Alfred Tarski. And in it he talks about ...
1
vote
2answers
101 views

Why is this a sentence with a quantifier an open sentence?

From these notes on Relational Logic, the following two sentences are given as examples of 1) open and 2) closed sentences. The definition of an open sentence is one with at least free variables. ...
0
votes
2answers
54 views

Constraint satisfaction problem - Arc consistency

The Constraint satisfaction problem (CSP) is roughly speaking a formalism that defines a finite set of relations over a domain. The relations are simply defined by enlisting elements in certain ...
7
votes
3answers
218 views

Why is the universal quantifier $\forall x \in A : P(x)$ defined as $\forall x (x \in A \implies P(x))$ using an implication?

And the same goes for the existential quantifier: $\exists x \in A : P(x) \; \Leftrightarrow \; \exists x (x \in A \wedge P(x))$. Why couldn’t it be: $\exists x \in A : P(x) \; \Leftrightarrow \; ...
5
votes
2answers
89 views

Is there a theory of extensible definitions?

We can define $+$ as a function $\mathbb{N}^2 \rightarrow \mathbb{N}$, and then prove: Theorem 1. The range of $+$ is $\mathbb{N}$. If we later wish to extend $+$ to a function $\mathbb{Z}^2 ...
5
votes
2answers
129 views

Does this qualify as a statement?

Is this a statement? All positive integers with negative squares are prime. What do we need to qualify as such?
1
vote
1answer
39 views

Are these two statements(theorems) equivalent?

I am given this theorem: Let $H$ be a check matrix for a linear code $C$. Then $C$ has minimum distance $d$ iff. there exists a set of $d$, but no set of $d-1$, linearly dependent columns in ...
1
vote
1answer
189 views

Meanings of the terms “conjunct” and “disjunct” in a logic?

This sentential logic problem is stated as: Suppose that $A \models B$, where $A$ is a conjunction of literals and $B$ is a disjunction of literals. Show that $ \models \neg A$, $ \models B$, or a ...
5
votes
4answers
4k views

Is the empty set a subset of itself?

Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty ...
1
vote
5answers
647 views

Definition Of Symmetric Difference

The definition of a symmetric difference of two sets, that my book provides, is: Set containing those elements in either $A$or $B$, but not in both $A$ and $B$. So, in set builder notation, I figured ...
11
votes
5answers
388 views

what is the definition of $=$?

what is the definition of $=$? Above is the question that I would like to be answered, below are some of my thoughts. I've been thinking about what it means to say $A = B$ I came to this from ...
8
votes
3answers
80 views

$(\Bbb R \to \Bbb R : x\mapsto x^2)\equiv(\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2) \not\equiv (\Bbb C \to \Bbb C:x\mapsto x^2)$

Consider the following functions: $f:\Bbb R \to \Bbb R : x\mapsto x^2$ $g:\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2$ $h:\Bbb C \to \Bbb C:x\mapsto x^2$ I'm quite sure that $h$ is not equal to $f$ ...
2
votes
2answers
113 views

The problem of bound variables in mathematical definitions

I was reading Paul Bernays’ Axiomatic Set Theory recently; in the book, Bernays gives the following definition of ‘ordinal number’. \begin{align} \text{On}(\alpha) \stackrel{\text{def}}{\iff} ...
0
votes
1answer
234 views

Definition of effective enumerability and empty set

Let $S$ be a set. We say that $S$ is effectively enumerable iff (by definition) there exists a function $f \colon N \to N$ which has $S$ as codomain. My question is: is the empty set an effectively ...
2
votes
1answer
240 views

Free boolean algebra

Consider the following definition: Let $X$ be a set and $e : X \mapsto A$ a mapping to a boolean algebra $A.$ We say that $A$ is free over $X$ (with respect to $e$) if for every mapping $f:X ...
2
votes
1answer
175 views

define simultaneous substitution recursively

Can you help me with my approach to the following task: Define simultaneous substitution $\phi[\psi_1,...,\psi_k/p_1,...,p_k]$ recursively. Usually we have recursive definitions about formulas, but ...
2
votes
3answers
249 views

A question about a certain way to define mathematical objects

It is common in mathematics to see definitions of the following form: we begin with a certain object $A$. we perform some construction depending on a choice of some parameter $\lambda\in\Lambda$ for ...
2
votes
1answer
253 views

Is Gödel's completeness theorem a representation theorem?

In general a representation theorem is — according to Wikipedia — a "theorem that states that every abstract structure with certain properties is isomorphic to a concrete structure". ...
5
votes
2answers
642 views

How to write “let” in symbolic logic

How do I write let in symbolic logic? For example, if I am in the middle of a proof and there is a variable which I can assign to an arbitrary value, what would I write? My best guess is: $$ x := a ...
6
votes
2answers
832 views

Precise definition of “weaker” and “stronger”?

If I say that $A$ is stronger than $B$, do I mean that $A \Rightarrow B$, or that $B \Rightarrow A$? (Or something else?) I feel like I have seen both usages in literature, which is confusing. ...
1
vote
1answer
213 views

Expressing P = NP as a first order formula

I want to express P = NP in a completely formal way. My first try: There exists an algorithm A and a polynomial bound p such that for all input i, A(i) = true iff i is a satisfiable formula and ...
4
votes
2answers
428 views

True, false, or meaningless?

Are the following two assertions always true, always false or meaningless? $\exists i \in \emptyset$ $\forall i \in \emptyset$ Because it seems that one encounters expressions of this kind fairly ...
2
votes
1answer
284 views

Set-theoretical definitions of the notion of “structure”

What general set-theoretical definitions of the notion of "structure" are there? By general definition of "structure" I mean a formula $\Phi(x)$ in the first-order language of set theory such that ...
1
vote
2answers
154 views

Informal Equivalents of Mathematica “Set” and “SetDelayed”

How would one distinguish between what is meant by Mathematica's "Set" and "SetDelayed" functions in informal mathematical notation? Is there a way to make this distinction any any reasonably standard ...
4
votes
1answer
107 views

Help on a definition

In many books I find these two definitions of Ramsey ultrafilters, extremely similar but different: 1)For every partition $\mathbb{N}=\bigsqcup A_k$ with $A_k\not\in\mathcal{U}$ there exists ...
4
votes
1answer
169 views

Definition of the union of structures?

Using the logic definition of a structure as a set coupled with finitary functions and relations, what is the definition of the union of two structures $\mathfrak{A}_1 \cup \mathfrak{A}_2$? I ...
8
votes
8answers
392 views

Is the 'variable' in 'let $y=f(x)$' free, bound, or neither?

Consider the string 'Let $y = f(x)$." Suppose that it occurs in some elementary context, such as when graphing the function $f$ using $x$/$y$ coordinates. How is this to be understood in predicate ...