Questions tagged [axioms]

For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.

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Showing Euclid's Proposition 30 ("Lines parallel to the same line are parallel to each other") is equivalent to the 5th Postulate.

I am trying to show that the 30th Euclid's proposition, "Straight lines parallel to the same straight line are also parallel to one another." is equivalent to the 5th Postulate: "If ...
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Is this theory trying to capture the theory of the minimal model of ZFC correctly formalized?

I'm trying to capture theory $T_0$ written by Noah's answer to a prior posting of mine. First we add a constant symbol $\mathcal M$ to the language of set theory. Now we add all axioms of $\sf ID$ and ...
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Is it valid/useful to prove statement $X$ by finding $Y$ such that $Y\to X$ and $\lnot Y\to X$?

Background: Suppose I want to prove theorem X. Typically, I'd have to use a set of axioms $A = \{A_1,A_2, \ldots ,A_n\}$ or previously proved theorems $T=\{T_1,T_2, \ldots,T_n\}$ and consider all of ...
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Meaning of notation S(x) in set theory

I am a beginner to set theory. axion of specification: To every set A and to every condition S(x) there corresponds a set B whose elements are exactly those element x of A for which S(x) holds. What ...
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Is $-\vec{v}$ in vector space axioms mean $-1$ multiplied with $\vec{v}$?

Is $-\mathbf{\vec{v}}$ in vector space axioms mean $-1$ multiplied with $\mathbf{\vec{v}}$? I got this doubt while i am solving the question below. Let $V$ be the set of positive real numbers ...
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Relationship between Axiomatic geometry and Hyperbolic geometry

I have done quite a bit of hyperbolic geometry and euclidean geometry, but one thing remained obscure to me throughout: the connection between axioms and the metrical definition of geometry. As I ...
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What is the rationale for selecting the value "one" as the probability of sample space?

Among the three axioms of the probability theory, the following is said to be the normalization axiom $$p(\Omega) = 1$$ It states that the probability of the entire sample space ($\Omega$) is equal to ...
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All possible proofs within an axiom system [duplicate]

I firmly believe people have asked this question, but I didn't manage to find it. So I post it again. I'm wondering, couldn't we make ever true statement within an axiom system that can be proven by ...
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How is circumscribing a triangle equivalent to the Parallel Postulate?

This question states that one of the statements equivalent to the parallel postulate (Euclid 5) is "Every triangle can be circumscribed". The Wikipedia page on Tarski's Axioms lists three ...
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Equality in category theory seems poorly defined to me

apologize if this doesn't make much sense, I am self-taught and often I am thinking about things completely wrong, but I am very lost right now. When we consider some generalization of an idea[^1], ...
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Why Hilbert's 24th Problem is unsolvable?

Hilbert's famous 24th problems handles the problem of the simplest possible proof of a mathematical statement, in a nutshell. It is said that there are a few problems with this problem. First of all, ...
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How to justify the necessity of the Axioms?

I am studying the logical fundaments of mathematics, but very often I have trouble to understand Peano's and ZFC/ZFC axioms. In Tao's book Analysis I, I found very helpful when he points out what ...
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Mathematical science being a tautology inquiry by Poincare

I was reading a quote from Poincare from his "On the Nature of Mathematical Reasoning" that states: THE very possibility of mathematical science seems an insoluble contradiction. If this ...
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Field definition - Unique inverse [duplicate]

In Abstract Algebra: Theory and Applications, every non-zero element in a field has an unique multiplicative inverse. However, on Wikipedia the definition has dropped the unique term. For example, is $...
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How is reductio ad absurdum unintuitive?

I am not asking about opinions here, I'm asking for the reasoning behind the decision of certain logics/frameworks, like e.g. Intuitionism, to not contain RAA as a valid rule of inference. There is an ...
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What different topological properties do spaces of different separation axioms have?

What different topological properties do spaces of different separation axioms have? I am looking for somehow "categorizing" topological (and non-topological) properties of spaces according ...
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5 answers
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What is the motivation for the axioms for Propositional Calculus in Mendelson's "Introduction to Mathematical Logic"?

On pp. 26-27 of his Introduction to Mathematical Logic (5th edition), Elliott Mendelson writes: If $\mathscr{B}$, $\mathscr{C}$, and $\mathscr{D}$ are wfs of $\mathrm{L}$, then the following are ...
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In ZFC, why can't $\{\{A\}\} \in A$? [duplicate]

The Axiom of Regularity prevents quine atoms like $A=\{A\}$ because the intersection between $A$ and $\{A\}$ is non-empty. However, I can't seem to find any contradiction with $A=\{\{A\}\}$ since the ...
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2 votes
1 answer
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Axiomatization of $(\mathbb Z, <)$

I'm interested in the axiomatization of the total order $(\mathbb Z, <)$. My idea is to have first the axioms for a total order: $\exists x : x = x$ $\forall x : \lnot(x < x)$ $\forall x : \...
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Consistent and independence of ZFC

My question is very simple but I want to make sure. when we say the statement is independent of ZFC means, we can not this statement true or false by using only the axioms of ZFC. Is that right? The ...
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Axiomatic derivation - what does instancing an axiom practically entail?

I'm stunned by the chapter in my coursebook (which is in Dutch, so please advice if I am mistranslating any of the terms) about deriving from a system of axioms and derivation rules. The exercise is ...
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If A is a Subset of B, then the closure of A is a Subset of the closure of B.

Conditions Hello, I am trying to prove that $A \subseteq B \implies CL(A) \subseteq CL(B)$. I know how to prove when you define closure of set E as the Intersection of all sets that are closed and ...
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Prove all 4 axioms of "less than" are necessary (for real numbers)

One way to define an ordered field is as a field $F$ with a relation $<$ that satisfies: For all $x,y \in F$, exactly one of $x<y$, $x=y$, $y<x$ holds. For all $x,y,z \in F$, if $x<y$ and ...
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Real Numbers Cannot be Constructed: Question about Constructive Mathematics

I got into a discussion with someone stemming from the set of uncomputatble numbers and how they claimed that such numbers like $\pi$ (not uncomputable but you'll see in a second) don't exist. I was ...
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Must axioms be proven to be compatible with other axioms?

As we know, an axiom cannot be proven. It is an assumption that certain properties exist, and then we can learn more interesting properties (theorems) from anything that obeys these axioms. So ...
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Could we use predicates instead of propositions in the definition of axiomatic system?

An axiomatic system is a finite sequence of propositions a_1,a_2..,a_N which are called axioms 56:23 In the whole lectures, two kind of logics are introduced: Proposition: A variable which is either ...
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3 answers
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Why is the Axiom of Pairing required?

I first heard of the ZF Aziom of Pairing watching this. I don't get why it is necessary to have an axiom which states that a set exists. Doesn't a set exist simply by virtue of the fact that it has a ...
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Formal proof that $\vdash (\varphi \rightarrow \psi) \rightarrow (\neg \varphi \lor \psi)$.

I have to prove the statement $$\vdash (\varphi \rightarrow \psi) \rightarrow (\neg \varphi \lor \psi)$$ only using first order logical axioms (similar to the ones in the Hilbert System), modus ponens ...
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Is it allowed that an axiom can be provable from the first order logic?

In "Elements of Set theory" by Enderton 54~55p We can form something like the Cartesian product of infinitely many sets, provided that the sets are suitably indexed. More specifically, let $...
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Proving $n\not\in n$ for $n\in\mathbb N$ without regularity

Is there a way to prove in ZF theory without regularity axiom $\forall n\in\mathbb N$, $n\not\in n$? At this point, I haven't proved yet that $\mathbb N$ is a well-ordered set, so a proof of the well-...
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2 votes
2 answers
132 views

How to find a 'set' after knowing the axioms of ZFC?

This was a question I had ever since I started studying Formal mathematics. Take ZFC for example, in it the axioms tell us 'tests' to check if something is a set or not and how the object, if they are ...
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1 answer
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A restriction on defining set in subset axiom

Subset axioms$\;\;\;$ For each formula $\varphi$ not containing $B$, the following is an axiom: $$\forall t_{1}\cdots\forall t_{k}\,\forall c\,\exists B\,\forall x(x\in B \iff x\in c \;\&\; \...
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2 answers
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Is conditional probability axiomatic? [closed]

Is the conditional probability $P(A|B) = P(A \cap B)/P(B), P(B) \ne 0$ an axiom in the Kolmogorov scheme or is it derived from the other axioms?
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Why lines and planes as primitive notions?

I'm preparing geometry classes and I thought it is a good time to answer a question I had when I started to study geometry: why, in Euclidean axiomatic geometry, is the notion of a straight line ...
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1 answer
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How to prove that if $a,b$ are positive reals then also $ab$? [closed]

It might be a silly question but I can't find how to prove this. I assume this has a proof per axiomatic system but I am looking for proof which is most standard, using the most common axiomatic ...
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Axioms for vector space and affine space under the same system and notation

Background: Here are common axiomatic definition of a vector space and an affine space: A vector space is a set together with two operations satisfying the following eight axioms. For addition: ...
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Is the Axiom of Extensionality needed in predicate logic without equality?

In predicate logic with equality, defining the Axiom of Extensionality allows us to gain the logical property of substitution. I have seen the axiom formulated as \begin{equation} \forall x: \...
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Does $-1×-1=1$ in any ordered field? [duplicate]

Let $F$ be a field equipped with the addition (+) operation and multiplication (×) operation with axioms https://mathworld.wolfram.com/FieldAxioms.html In general does $-1×-1=1$? In what cases would ...
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6 votes
1 answer
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What exactly is the distinction between a theory and model in model theory if models are themselves constructed in axiomatic theories?

My understanding is that model theory requires a distinction between a logical theory and a structure to interpret the statements of the theory. However, every piece of mathematics including every ...
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Scope of Quantified Variables in the Axiom of Separation in ZF

I realize in the axiom of separation $$x\in A \leftrightarrow \phi(x)\wedge x\in B$$ the free variable $x$ is restricted to $B$ to avoid Russell's Paradox, but are the quantified variables in $\phi$ ...
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Confused about the consistency of ZFC

Unfortunately I have never taken a class in mathematical logic. Shame on me. But I am reading here (https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC) that the consistency of the ZFC ...
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3 votes
2 answers
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Vector spaces without additive inverses

I was writing out the axioms of a vector space, in preparation for teaching next week, and I started wondering: Do I actually need to impose that vectors have additive inverses? To be precise: Let $(F,...
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Axioms for a Neighborhoods Motivation

On a topological space $X$, the class of all neighborhoods of $x\in X$ formalizes the intuitive concept of "closeness" to the point $x$. These sets are usually defined using open sets on $X$...
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Is the Axiom of choice intuitive? How was it first introduced?

I will refer to the Axiom of choice as ($AC$). As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms together with the $AC$. But the last axiom seems to be the most ...
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Rule enforcing consistency of a logical set

The principle of explosion states that a single contradictory statement among a logical set is enough to prove all propositions wrong. So, for a set to be considered valid, it must be consistent*. ...
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4 votes
2 answers
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If the axiom of replacement implies the axiom of specification, why are both mentioned

In Tao's Book on Analysis I, we are asked to prove that the axiom of replacement implies the axiom of specification. This implication seems to be true even outside of the environment Tao sets up in ...
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Constraints on the variables used in the Axiom of Extensionality formal definition?

The formal definition of the Axiom of Extensionality in ZF is: $$ \forall x \forall y \ (\ \forall z \ (z \in x \iff z \in y ) \implies x=y\ ) $$ I am having trouble interpreting this particular ...
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3 votes
1 answer
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Interesting examples of first-order, one-sorted proper extensions of PA

Are there any interesting nontrivial examples of first-order, one-sorted theories of natural numbers (i.e. theories whose quantifiers range over natural numbers only and not, say, sets of natural ...
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In the book by Apostol "calculus volume 1" how to prove that sum of two integers is an integer?

In Apostol's book we start by defining a set called the set of real numbers which satisfies the field and order axioms. Then we define the set of positive integers as being the subset of every ...
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3 votes
1 answer
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How do model theorists define structures?

Let $L$ be a first order language and $A$ be an L-structure. Let $\sigma$ be an L-sentence. In most of the mathematics I have seen a structure is defined by the axioms it satisfies. These may or may ...
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