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Questions tagged [axioms]

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

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158 views

Bases in vector spaces without $AC$

It is known that without the axiom of choice, not every vector space has a basis. But I was wondering, if I don't assume the axiom of choice, and I choose a vector space $V$ which does have a basis (...
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163 views

Why are the laws of sines/cosines “laws” and not “theorems”?

So in logic we have every line of a proof being either an axiom or a theorem -- but then why do we have concepts like the "law of sines" and the "law of cosines"? Are these technically "theorems" as ...
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107 views

Does introducing a new type create a different extension, and is it still conservative?

In this comment on Terry Tao's page about his Analysis I textbook, he writes, If one wanted to do things by the book, what one should actually do each time one introduces a new mathematical object, ...
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116 views

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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89 views

Multiplying both sides of an equation in proofs

I'm learning the basics of group theory, and must justify every step of a proof by referring to the basic axioms/theorems. Which axioms/theorems justify multiplying or adding an element of a group to ...
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378 views

Geometric basis for the real numbers

I am aware of the standard method of summoning the real numbers into existence -- by considering limits of convergent sequences of quotients. But I never actually think of real numbers in this way. I ...
5
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210 views

Neighborhood Topology and Open Set Topology: their Equivalence and Comparison

Motivation. A few years ago we were using Armstrong's Basic Topology as a textbook for the topology course in my university, and right off the bat I had a huge conceptual problem regarding the two ...
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174 views

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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73 views

When is the (great) axiom of Union really needed

Consider these two (informally stated) axioms: (Small Axiom of Union) For any two sets $A,B$ there exists the set $A \cup B$. (Great Axiom of Union) For any set $A$ there exists its union $\bigcup A$....
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159 views

How to start with $\pi$ defined as the area of the unit circle.

In IV-1, Example 3 of Advanced Calculus of Several Variables, by C. E. Edwards Jr. the reader is told "Since $\pi$ is by definition the area of the unit circle,...". The example provides a proof that ...
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320 views

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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419 views

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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90 views

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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35 views

An introduction book for analysis with the axioms of Tarski

Briefly: Is there an introduction book for analysis with The axioms of Tarski as basis? (I found them very elegant. And the most important thing for me is that the axioms are "obvious", not like the ...
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64 views

Is every proposition on Cartesian geometry provable on synthetic Euclidean geometry?

Obviously everything that is associated with coordinates can’t be analyzed within synthetic geometry. But existence, measure and incidence statements are provable; since Cartesian geometry is an ...
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64 views

Is there an axiomatic characterization of the Lebesgue integral?

Is there an axiomatic characterization of the Lebesgue integral w.r.t. some finite measure $\mu:\mathcal{F}\rightarrow[0,\infty)$, for instance as the function $I$ over the set of real-valued, $\...
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99 views

How to expand second-order ZFC to include classes?

The system of second-order $ZFC$, presented in Shapiro, "Foundations without Foundationalism", is formulated in second-order logic and includes the usual axioms of extensionality, foundation, pairs, ...
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45 views

Equivalence between different formulations of homology axioms

For a reduced (co-)homology theory defined for the CW-complexes, is the following formulation equivalent to the axioms of excision and long exact sequence? It states that for a CW-pair $(X,A)$ there ...
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291 views

Induction Implies Well Ordering

Every proof that induction implies well ordering I have seen goes: assume $S\subset\mathbb{N}$ has no least element and let $T$ be its complement with respect to $\mathbb{N}$. Since $1$ is the ...
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64 views

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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364 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 ...
3
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59 views

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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68 views

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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44 views

Axiomatisation of Infinite Series

I have seen a ridiculous "proof" claiming that $$\displaystyle\sum_{n=1}^{\infty} n=-1/12.$$ The starting point is a false statement that $$\sum_{n=1}^{\infty} (-1)^{n+1}=1/2.$$ Since we know that ...
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76 views

theories where angles exist without a metric

(moved from https://mathoverflow.net/questions/307703/theories-where-angles-exist-without-a-metric) The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible ...
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86 views

Minimal axioms necessary to prove the incompleteness theorems?

What's the minimal sufficient (plausibly) consistent system of axioms to prove the First incompleteness theorem? More interestingly can the First incompleteness theorem be proved in a consistent self-...
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230 views

Shortest distance between two points is a line proof

Two questions (please bear with my curiosity as I am still somewhat a beginner) - is this considered to be a theorem or just an axiom? In general, how does one come to know if a statement is an ...
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160 views

Is it possible to prove the zero product property in a vector space the other way around.

EDIT: problem solved don't bother reading. In a vector space $V = (V, F)$, the zero product property states that for all $\lambda \in F$ and $\underline{v} \in V$, if $\lambda \cdot \underline{v} = \...
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41 views

Existence of a Precise axiomatization of Eudoxus theory of magnitude

Is there a precise axiomatization of the Eudoxus theory of proportions? For example, a) (D, +, <) is a structure such that < is a strict linear order, b) + is an order-preserving ...
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75 views

Is an equivalence relation (= sign) needed for the real number system or is a consequence of the other axioms?

My math education is based on Calculus or Real Analysis didactical books intended for bachelor's degrees, mainly read in chunks, and never went any further. Generally, the Real Number System is said ...
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45 views

Tidy Collection of Axioms Used in Mathematics

The power of mathematics relies on its logical representation and it depends on its strict disjunction between axiom and definition. I'd learned lots of definitions while studying mathematics, but ...
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22 views

How quickly can EFA define things, asymptotically?

EFA is theory of arithmetic. For a number $l$, we define $EB(l)$ as the largest $n$ such that there is a predicate $\phi$ with $l$ or less symbols such that $EFA \vdash \forall x. \phi(x) \iff x = n$,...
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76 views

Origami Axiom 4 is Redundant? (Huzita–Hatori)

I have some question regarding Huzita–Hatori axioms. Axiom 4 states that given a line and a point, we can make a fold passing through the point perpendicular to the line. Axiom 5 states that given ...
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46 views

Why is the axiom of foundation/regularity called “regularity”?

I'm curious about the history of the word "regularity" in the axiom of foundation/regularity. Both of those terms are used interchangeably (Wikipedia and most texts seem to use "regularity", while ...
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38 views

Logarithm completeness axiom proof

Consider the set $$ S = \{ r\in \mathbb{Q} \mid 2^r < 3 \}$$ Our professor wants us to show a few things, 1: that S is bounded from above. 2: if $ 2^\delta < 3 $ there is an $n \in \...
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109 views

Shelah Soifer graph and the axiom of choice

Define the Shelah-Soifer graph $G= (\mathbb{R}, V)$ as $\{x, y\} \in V \iff \exists r\in \mathbb{Q}, \epsilon \in \{1,-1\}, x-y = r + \epsilon \sqrt{2}$. First of all, one notices that if $x$ is any ...
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71 views

A question on Axiom XI of Veblen's paper on the axioms of geometry

Recently I have started reading Oswald Veblen's A System of Axioms for Geometry. There it is written that (see page 346), Axiom XI. If there exists an infinitude of points, there exists a certain ...
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206 views

Ordinal Numbers within Simple Type Theory

A little preamble: I recently came across the system of higher-order logic known as simple type theory, and I was immediately intrigued by it, as it seemed to be exactly what I was looking for as an ...
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191 views

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, exactly?...
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278 views

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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238 views

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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239 views

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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26 views

Axiomatize inner product spaces in terms of angles?

Inner product space is a vector space $V$ over a field $F$ together with an inner product $P:V\times V\to F$ that satisfies the inner product axioms. This inner product induces an angle $\angle (v,w) ...
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44 views

Were there any results that were true in old axioms, but are false in modern axioms, and are not obvious paradoxes?

Recently I learned that it can't be proved that mathematical axioms are consistent. And furthermore, in 1900s math was based on an inconsistent system of axioms. So, are there any results, that were ...
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25 views

How to determine if an axiom is consistent, independent, complete, and/or categorical

An axiomatic system is: consistent: if no logical contradiction can be derived from the axioms. (Don't see how you can prove there is no logical contradiction possible.) independent: if an axiom ...
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48 views

Incompleteness of Robinson's Arithmetic

In Smith's book about Godel's Theorems a proof is given of incompleteness of Robinson Arithmetic. It's done by producing a special interpretation of the theory: Domain is $N^* = N \cup \{a,b\}$, a ...
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25 views

Is there a formal connection between transitivity and compositionality?

Transitivity is the property of a relation: $R_{ab}\land R_{bc} \to R_{ac}$. Compositionality is an operation on functions: $\circ : (A\to B) \times (B \to C) \to (A \to C)$. Intuitively, these are ...
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31 views

Is there correspondence between vector spaces and independent theories?

I'm not a mathematician, but I have this idea. A vector space is defined by a basis, that is a set of linearly independent vectors. On the other hand, an independent theory is defined by a set of ...
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46 views

Fundamental axiom or theorem for multiplication of an equation by any real number?

Is there a fundamental axiom or theorem stating that if two quantities are equal, multiplying both quantities by the same scalar real number results in two equal quantities? I'm imagining something ...
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66 views

Is there a finite axiomatization of Tarski's geometry axioms?

Tarski's geometry axioms include an axiom schema, the axiom schema of continuity. Let $\phi(x)$ be a first order formula not containing $y$, $a$, or $b$ as free variables. Let $\psi(y)$ be a first ...