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

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Definition: Mathematical way to define the “left” and the “right”

How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has ...
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Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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1answer
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What does it mean to axiomatize a logic?

I'm sorry if this question is not clearly formulated: An axiomatization, or an axiomatic system, usually means a set of axioms (i.e. a theory). A formal theory is such a set of formulas in some formal ...
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80 views

How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
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Alternative Axiomatic Systems

At least as I understand the motivation behind rigorous definitions of the foundations of mathematics (the only contender with which I'm familiar being ZFC and extensions), the idea of an axiomatic ...
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Vector space and axioms

How would i go around proving a(u+v)+ u= (a+1)u+av using axioms? I started with distributivity with respect to vector addition, associativity of addition, commutativity of addition, distributivity ...
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42 views

How do I prove this implication using axiomatic methods?

I am trying to prove that $a >0$ if and only if $\frac{1}{a} >0$, But I am having a lot of trouble doing so. So I would have to prove both sides of the implication. That is $a>0 \implies ...
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How to establish the formula for area of a triangle using the axioms of area?

We have the following definition: AXIOMATIC DEFINITION OF AREA We assume there exists a class of $M$ of measurable sets in the plane (i.e. subsets of the plane whose area can be defined) and a set ...
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How to know when a system of axioms is 'complete'?

Here, I (basically) stated the group axioms as follows. $(xy)z=x(yz)$ $xe=x, ex=x$ $xx^{-1}=e$ In that post, answerers Martin and Ittay were critical of the above list for not including ...
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Axiomatizing topology through continuous maps

Suppose we have some topological space $X$ and we somehow forgot about the topology. A friend of ours knows the topology and offers to tell us for any map $X\to Y$ into any topological space $Y$ ...
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What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
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Prove that (M,+,*) is a field

Prove that the multiplication $*:M \times M \to M$ defined by this table: * | 0 1 -------- 0 | 0 0 1 | 0 1 together with the commutative group (M,+), is a field (M,+,*). Group axioms: 1) ...
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Axioms of Newtonian Mechanics

Axiomatically speaking, could Newton's laws be derived (as theorems) from the conservation of momentum and energy -- along with a few suitable definitions of things like an inertia frame and force? ...
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Axioms as recreational mathematics

Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the ...
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Has anyone proposed an axiom for ZFC that implies that many different cardinal characteristics are distinct?

I think it is more interesting when cardinal numbers are distinct, rather than equal. Has anyone proposed an axiom for ZFC that, in one fell swoop, implies that many different cardinal characteristics ...
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Strength of “Every finite dimensional subspace of a vector space has a complement”

Does the following choice principle have a name? Every finite dimensional subspace of a vector space has a complement. Equivalently, every line inside a vector space has a complementary ...
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Proving Boolean Logic using Axioms

I am trying to prove boolean logic formulas using axioms. I have a lot of trouble deciding what to do, which axiom to use, when to use it, etc. I've asked others on proving formulas but they have all ...
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195 views

Power-set in Hypercube: historical background of indexing each term like Hasse Diagram?

My instructor wants references about the indexation over the hypercube, related question here, he does not believe that I was the first who used it -- [update] thanks to a comment, the name is Hasse ...
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Have people studied weakened forms of Tarski's axiom?

My intuition about Grothendieck universes says the following. Axiom of empty set $\leftrightarrow$ There exists a Grothendieck universe. Axiom of infinity $\leftrightarrow$ There exist at least two ...
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A mathematical object that can be characterised by axioms as well as by universal constructions

This question (more specifically one of the comments) prompted me to ask this question: Can someone give me an example of a mathematical object that one can characterize both via axioms and universal ...
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Dependence of axioms of Zermelo set theory

I'm reading a book about Zermelo–Fraenkel set theory, and I was told that there are a lot of dependence relation among the commonly used axiom system (in the question What axioms does ZF have, ...
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Are there any formal theories that are not axiomatic?

Are there any formal theories that are not axiomatic? Could you give me some examples?
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Simple Proofs in ZFC Set Theory

So I'll keep this real short and simple. In this document on page 23 there is a list of axioms. On page 24 there is a list of theorems that come from said axioms. I can prove them all except 3 and 5. ...
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Problem solution by model theory

Sorry if that's not the right place for asking this, but didn't have anywhere else to go. I was cheking out some math problems in the Mathematical Olympiad site, and I found this one: Let $\mathbb ...
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Lebesgue Measurable Sets & Axiom of Determinacy

While reading some logic theory I bumped against the theorem which states that every set of reals is Lebesgue measurable, assuming the axiom of determinacy. To prove this theorem it apparently ...
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with Tarki's axioms of geometry what is a plane?

Reading about Tarski's axioms of geometry https://en.wikipedia.org/wiki/Tarski%27s_axioms I was puzzeling how they would look for 3 dimensional geometry. the big problem then is the upper dimension ...
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Huzita Axiom 6 - Computing the Origami Trisection of an Angle

The Galois theory proof of the improssiblity of angle trisection rests on studying the triple angle formula $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$. Ruler and compass numbers can only be ...
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Axiomatisability of the class of finite groups

I'm trying to prove that the class of all finite groups is not finitely axiomatizable. I was trying to do this in the same way one proves that the class of finite sets is not axiomatizable, i.e. Let ...
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Von Neumann universe implies Foundation

Suppose we are working in $\mathsf{ZF-Foundation}$. Additionally, let us assume that $V = \bigcup\{V_{\alpha} | \alpha \in \mathsf{Ord}\}$ holds. Now I want to show that this implies Foundation. I am ...
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Are there any articles (or otherwise) that discuss the idea that the set-theoretic universe should be as “free” as possible?

Question. Are there any articles (or otherwise) that discuss (and perhaps attempt to formalize) the idea that the set-theoretic universe should be as "free" as possible? Discussion. ...
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Is there an axiom for ZFC that totally orders Cichon's diagram without collapsing it?

Is there a known axiom for ZFC that: Fixes a total ordering on the cardinal numbers appearing in Cichon's diagram. Is "natural looking" - in particular, its not allowed to be a conjunction of ...
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Is “autonym” an autonym?

I believe this is set theory, but I don't really know set theory, so... An autonym is a word that refers to a property that the word itself happens to possess. For example "noun" is an autonym ...
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Congruent figures have equal areas. Why?

When we are first introduced to the idea of congruent polygons (generally triangles) in school, we often define congruent figures as "figures which have the same shape and size". However, today, ...
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Set of true statements generated by set of axioms with a binary operator

I wondered about this and I am having a hard time formulating it as a question at all, but I hope I can express something if my wondering here. Assume we have a set $V = \{\mathbb N, +, =, X ,(,)\}$. ...
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Hilbert projection theorem without countable choice

All the proofs of the Hilbert projection theorem, existence part, that I have seen so far use countable choice (usually implicitly). Is this necessary? It seems like you might be able to leverage the ...
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Transitive property of equality and the fundamental nature of algrebra

The fundamental nature of algebra rests on the basic rule that whenever two numbers, variables, or expressions are equal, either one can be replaced at any time by the other one. For example, if we ...
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Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there ...
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Is a “model” only a proper model if everything in it's definition is also explicitly constructed?

Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the ...
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Using definitions instead of axioms.

Lets take (classical) first-order logic for granted, including an equality symbol and its associated axioms. Given all this, a rigorous work of mathematics will typically begin with a signature - ...
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Should every line be infinite in both two directions?

Yesterday one my classmate asked me for the the definition of line. I have found it in coordinate geometry, in vector space, and in metric space. The definition in metric space is quite general ...
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Order Axioms: '<' or '$\leq$'?

In the axioms given for the real numbers, I see that the order axioms are sometimes given for the '<' relation and sometimes for '$\leq$'. Which is more commonly used these days?
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Proving the distributive law in Real numbers

Given that the distributive law holds for positive real numbers a,b,c meaning that a(b+c)=ab+ac How do we extend the proof to all real numbers?
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Consistency in Related Sets of Axioms?

If I have a set of axioms A = {A1, ..., An} and if I create a set of axioms B = {A1, ..., An, Con(A)}, would it be true to say that Con(A) iff Con(B)? Is there a simple counter-example to this? More ...
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Is this proof correct using just the field axioms?

I have to show, using just the field axioms for the rational numbers, that $$(x+y)\cdot (x−y)=x \cdot x − y\cdot y$$ So I have started by: Replacing $(x + y)$ with a $k$, since addition is closed ...
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Axioms for substructures of an algebra.

Given an algebra $ A $ and subset $ S $ (EG. quaternion algebra H and subset $S=\{a+bi : a,b \in R\} $) What are the axioms to show check for $ S $ is a: subalgebra linear subspace (sub vector ...
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Pasch's axiom in the hyperbolic plane

Q: I am asked to show that pasch's axiom holds true in the hyperbolic plane. Pasch's axiom Idea: I was thinking of about a circle $\Gamma$ whose centre is the origin of the complex plane. ...
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Axioms for ordered fields

If i have a set of numbers, that contains every positive number, including 0 , except the negative ones, can i claim it as an ordered field? I would say no, because you can't find an $y$ for any $x$ ...
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Alternative definition of a matroid?

$\mathcal M\,$ is a nonempty subset of the set of all subsets of the finite set $X$, such that: $(1)\quad\; \neg\exists U,V\in\mathcal M:U\subset V\quad$ $(2)\quad\; A\subseteq B\in\mathcal M\wedge ...
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Quotient respect to an equivalence relation.

I have the natural numbers $\mathbb{N}$, in ZF. I want to construct the integers $\mathbb{Z}$ taking the quotient respect to usual the equivalence relation $R$ on $\mathbb{N}\times\mathbb{N}$ that is ...
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How to translate these defitions of bipartiteness to each other?

Let $(V,E)$ be a bipartite graph. I'm trying to capture this property and I've come up with two definitions and am surprised/confused that one uses a negation and the other doesn't - still I can't ...