# Questions tagged [axioms]

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

170 questions
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...