What is Axioms: Definition and 189 Discussions

An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning.As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic).
When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there may be multiple ways to axiomatize a given mathematical domain.
Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.

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  1. P

    What are the axioms in ZFC set theory?

    since a lot of talking is going on with sets, will somebody write down the axioms in ZFC theory as a point of reference , when a discussion is opened up. thanx
  2. M

    Linear Algebra-Fields and axioms

    Homework Statement Let F be any field, and fix a є F. Equip the set V = F2 with two operations as follows. Define addition by (x, y)‡(x', y') := (x + x', y + y' − a), for all x, x', y, y' є F, and define the scalar multiplication by scalars by c * (x, y) := (cx, cy − ac + a), for all x...
  3. S

    Geometry - Axioms and Point Relations

    Homework Statement 1. Suppose A * B * C and A * C * D. a) Prove that no two of A, B, C, D are all equal. b)Prove that A, B, C, D are all on one line. 2. Suppose that A, B, C are points not all on one line. Prove that AB and BC have no points in common except B. Homework Equations...
  4. A

    Decidability of -1<1/0<1 using the ordered field axioms and first order logic

    Is the statement -1<1/0<1 decidable using the ordered field/real number axioms and first order logic? I have tried to prove that the statement is either true or false but have had no success since the axioms and theorems only make statements about objects that exist and do not give any clear way...
  5. H

    Exploring the Axioms of Physics

    can physics be axiomed :confused:
  6. P

    Vector Space Axioms: 4 Rules to Redefine

    I am trying to shorten and generalize the the definition of a vector space to redefine it in such a way that only four axioms are required. The axioms must hold for all vectors u, v and w are in V and all scalars c and d. I believe the four would be: 1. u + v is in V, 2. u + 0 = u 3. u...
  7. D

    Which axioms of ZF are needed for Cantor's theorem?

    Self explanatory. The Cantor's theorem which am referring to is that the cardinality of the power set of any set is greater than that of the set.
  8. MathematicalPhysicist

    Examples for seperation axioms.

    My assignment is like this: 1.give an example of a space X and a subspace A of X s.t X satisifes Sep and A doesnt. 2.give an example of a continuous and onto function f:X->Y s.t X satisifies S1 but Y doesnt. 3.give an example of a continuous and onto function f:X->Y s.t X satisfies S2 and Y...
  9. Fredrik

    Mathematica Mathematical theories and axioms

    Maybe this is a dumb question. I'm a bit tired right now. :smile: What is a "theory" in mathematics, and what kind of statements can we call "axioms"? To be more specific, is "group theory" a mathematical theory, and if yes, what are its axioms? Should I think of the definition of "group"...
  10. T

    Proving 0v=0 using only the 10 Axioms

    Well, I'm supposed to prove 0v=0 It is stated that I'm only allowed to use the following axioms. let a,b,c be vectors and V is a vector space, then 1)a&b is in V then a+b is in V 2)a+b=b+a 3)a+(b+c)=(a+b)+c 4)0+a=a+0=a 5)a+(-a)=(-a)+a=0 6)a is in V implies ka is in V...
  11. A

    Prove a few theorems about vector spaces using the axioms

    Hey guys, I need to prove a few theorems about vector spaces using the axioms. a) Prove: if -v = v, then v = 0 b) Prove: (-r)v = -(rv) c) Prove: r(-v) = -(rv) d) Prove: v - (-w) = v + w where r is a scalar and v, w are vectors.
  12. P

    What are the axioms that all of mathematics is built from?

    I assume someone has figured this out... If so, would anyone mind explaining it to me?
  13. E

    Using one-sided axioms to show <G,*> is a group

    Homework Statement Prove that a set G, together with a binary operation * on G, satisfying the following three axioms: A1) The binary operation * on G is associative A2) There exists a left identity element e in G such that e*x=x for all x in G A3) For each a in G, there exists a left...
  14. A

    Do mathematicians believe in axioms..

    :rolleyes: Do a mathematician believe in AXIOM like some people believe in GOD People who believe in GOD need not to proof GOD, is that like the way mathematician do with AXIOM ?
  15. K

    How do axioms hinder scientific progress and benefit society?

    Axioms, as long as they remain axioms: 1) Are not falsifiable by experiment 2) Are not verifiable by experiment 3) When required by theories, are subject to questioning by non-scientists and conspiracy theorists 4) Make theoretical scientific notions which need them speculative rather than...
  16. P

    Linear Algebra: Vector space axioms

    Homework Statement One of the fundamental axioms that must hold true for a set of elements to be considered a vector space is as follows: 1*x = x I was given a particular space: The set of all polynomials of degree greater than or equal to three, and zero, and asked to evaluate whether or...
  17. quasar987

    Is Axiom 2 of Sigma Algebras Equivalent to X Being in the Sigma Field?

    Wiki says that a sigma algebra (or sigma field) is a subset \Sigma of the powerset of some set X satisfying the following axioms 1) E\in \Sigma \Rightarrow E^c \in \Sigma 2) E_i \in \Sigma \ \ \forall i \in I \Rightarrow \bigcup_{i\in I}E_i \in \Sigma (where the index set I is countable) Am...
  18. Y

    Math Axioms Problem: Solve Questions & Get Hints

    Hello, Can you please help me with the questions listed below. I would like to get hints on how I can solve them. I have listed first the axioms and then the questions at the bottom. Axioms: --------------------------------------------------------- A plane consists of: -two sets P...
  19. E

    How to Prove that (Z,+) is a Group?

    It is known that "the integers under addition" form a group, that is (Z,+). I have always wondered how to actually proof that (Z,+) is a group? Definitions for a group from wikipedia: http://en.wikipedia.org/wiki/Group_(mathematics)#Basic_definitions I'm especially interested in two things...
  20. M

    Where can I find a list of physics axioms?

    Are basic principles like those of Newton's laws of motion considered as physics axioms on which you expand to physics theorems? Where can I find a list of physics axioms?
  21. J

    Everyday I take axioms for granted

    Another thread got me thinking... Everyday I take axioms for granted, eg. muliplication, addition, ordering of reals. From the pure point of view, what axioms are the most important (most used) ones? Wikipedia has a list: http://en.wikipedia.org/wiki/List_of_axioms However, I'd like...
  22. D

    Axioms - the choice & meaning

    Hi! I'm a high school student and I've been interested in Logic for some time... Although I read some books and acquired some knowledge, I still have one question that remains unanswered in spite of my hard work... My tutor told me that the three axioms of Propositional Logic (see them for...
  23. B

    Exploring the Group Axioms: What Makes Them an Axiom?

    Hi. I'm reading a simple introduction to groups. A group is said to be a set satisfying the following axioms (called the 'group axioms'): 1) Associativity. 2) There is a neutral element. 3) Every element has an inverse element. 4) Closure. My questions is simply: why are they...
  24. S

    What are the axioms of Classical Physics?

    I've read several QM texts which list five or four axioms for Qm, from which the rest is derived. I was wondering what might the axioms of Classical Physics be. I'm assuming one of them is: \delta{S} = 0 What might the others be?
  25. R

    Proof using the Axioms of Addition and Multiplication

    So does everybody know the Axioms of Addition and Multiplication? They are too long to type, but they are listed: A1, A2, A3, A4, A5, M1, M2, M3, M4, M5 and the distributive law, DL. anyways, I want to prove: 1. (-x)y = -(xy) and 2. (-x)(-y) = (xy) using ONLY the axioms of additon and...
  26. G

    Vector Space Axioms which fail certain matrices

    Second week in Linear Algebra... My homework involves of identifying all failing Vector Space Axioms for various sets of vector spaces. I did fine with a "regular" set like (x,y,z) which has an operation like k(x,y,z)=(kx,y,z). I have worked through all 10 of the axioms, comparing left sides...
  27. MathematicalPhysicist

    Can we determine the number of theorems that can be proven with X axioms?

    X axioms... lets assume we have a consistent system which have X axioms/definitions, can we infer (or deduce) somehow from this given, how many theorems/lemmas are there in this system? (or deduce the maximal theorems/lemmas that could be proved in this this system). my initial answer would...
  28. H

    Separation axioms vs subspaces

    How do separation axioms carry over to subspaces? Some are clear -- it's easy to see that if any two points of a space X are separated by neighborhoods, then the same must be true of any subset S of X. But what about the nicer ones? Is it true that if S is a subset of a normal space, that...
  29. E

    Probability Axioms: Deriving Version 2 from Version 1

    In my probability class we were given two versions of probability axioms which are equivalent. Let S be the sure event, A and B any arbitrary events, I the impossible event. I will use u to denote union, and n to denote intersection: Version 1 1. P(S)=1 2. P(A)>=0 3. If AnB=I, than...
  30. R

    Can Axioms Prove Falsity in Formal Axiomatic Theories?

    Hello. I am working on a paper detailing Godel's Incompleteness Theorm and I came across this statement. "An axiomatic base where all the axioms are true cannot prove anything to be false." Is this correct?
  31. honestrosewater

    Field axioms with or without closure

    I'm not studying algebra yet, I just happened to notice this and am curious. Mathworld's entry for the field axioms doesn't include closure axioms, but I have seen other authors include closure axioms in the field axioms. Does anyone know why this is or what difference it makes? Can closure be...
  32. S

    Question about Axioms and Theorems

    Okey, this might be a silly question. I know that theorems are deduced logically from the axioms. But I was just wondering is it possible to deduce an axiom from the theorems? In another words work backward, assuming the required theorems are known.
  33. E

    Simple proof, just using the axioms

    How could I show that (-m)(-n)= mn? The only thing I am allowed to use to prove this are the 5 basic mathematical axioms which allow for the commutative property and associative propery of the binary operations multiplication and addition;there exists an additive inverse for each integer, 1 is...
  34. marcus

    What is normally meant by quantizing a classical theory, versus Hardy's axioms

    Loop and Paden have brought up Hardy's axioms of quantum theory what do you think is usually meant by quantizing a classical (non-quantum) theory? And how does this connect to these axioms of what a quantum theory ought to be Here is a mainstream summary description of what quantizing...
  35. MathematicalPhysicist

    Exploring Hardy's Axioms and Their Impact on Quantum Theory

    what are they ? i know they are related to quantum theory.
  36. S

    Defining S: Axioms & Theorems

    A mathematical system S is defined as S={E,O,A}. A is the set of axioms describing the system. Is the definition of E considered an axiom? For example, if I want E={a,b}, then in the set A, do I write A={...,E={a,b},...}? Also, is the definition of O an axiom? Say O={~,V} and then I define...
  37. S

    Understanding "Sentential" Logic: Axioms, Notation & Binary Operations

    I came across this site: http://mally.stanford.edu/tutorial/sentential.html It lists four axioms of "sentential" logic. I first would like to know if there are other axioms not listed here. Wouldn't you need some axiom like if P is true, then ~P is false? It seems difficult to prove the law...
  38. MathematicalPhysicist

    Axioms of category theory

    what are them?
  39. MathematicalPhysicist

    Hilbert's 20 axioms of the Euclidean geometry

    what are they?
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