Tagged Questions

1
vote
1answer
54 views

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

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
63 views

Is there a theory of extensible definitions?

We can define $+$ as a function $\mathbb{N}^2 \rightarrow \mathbb{N}$, and then prove the theorem: Thm. The range of $+$ is $\mathbb{N}$. If we later wish to extend $+$ to a function $\mathbb{Z}^2 ...
5
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2answers
83 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
31 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
39 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 ...
-6
votes
3answers
157 views

Why is 0=0+0 not used as definition of 0? [closed]

I see my Ph D in philosophy uses ZF or so to mean that 0 is defined as the empty set. I disagree since 0 should mean false and absense of falsehood should not be defined as false. Why not simply ...
5
votes
4answers
184 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
109 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 ...
10
votes
5answers
314 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
73 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
82 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
144 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
170 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
123 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
181 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 ...
3
votes
1answer
199 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". ...
3
votes
2answers
329 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 ...
5
votes
2answers
362 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
196 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 ...
3
votes
2answers
338 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
267 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
143 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
95 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 ...
3
votes
1answer
157 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 ...
7
votes
8answers
362 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 ...
2
votes
1answer
112 views

does the definition of model depend on a theory or just a signature?

Σ:signature T:Σ-theory For M:Σ-model which satisfies T, is it proper to name it “Σ,T-model” or “Σ-model satisfying T”? In comparison, in the context of Lawvere theories, any Lawvere theory L ...
9
votes
2answers
362 views

Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials

[disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below] in my attempt to find a precise version of the 'definitions' usually given when ...