Axioms Definition and 189 Threads

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. caffeinemachine

    MHB Hilbert's axioms. Betweenness and ordered fields.

    I am reading the book "Geometry: Euclid and Beyond - Robin Hartshorne". Here's the first half of Proposition 15.3 from the book. If $F$ is a field, and if there is a notion of betweenness in the Cartesian plane $\Pi_F$ satisfying Hilbert's axioms (B1)-(B4), then $F$ must be an ordered field...
  2. H

    Proving \neg x \vee x in Hilbert System: A Logical Dilemma

    Homework Statement Prove \neg x \vee x using Hilbert system. Homework Equations The logical axioms. I'm not sure if I should state them, or whether there is a standard set. It seems to me that different sets are used. Anyway, the ones with disjunction in them are: a \rightarrow a \vee b...
  3. L

    Proving Every Line is Contained by Two Planes: Using Incidence Axioms

    I need to prove that every line is contained by at least two planes using only the incidence axioms. This is what I have so far... Conclusions Justifications 1. Let l be any line. Given 2. l has at least two...
  4. A

    Proving v=w using Vector Space Axioms

    Homework Statement V is an arbitrary vector space and v,w,x are part of V such that v + x = w = x Use vector space axioms to prove v = w I've looked at the axioms for an hour and can not get any lead to start this question.
  5. L

    Real Numbers: Axioms or Theorem?

    Hi there, In most books that I saw, the set of real numbers under the usual sum and product is considered as a Field and say that's by the field axioms. But I have surprised when I have seen, it is a theorem. The question, are these axioms? or can they be proved? Thank you very much.
  6. I

    Apostol's Field Axioms Homework: Solving Problems and Proving Theorems

    Homework Statement I've been steadily working through Apostol's Calculus, Volume 1 and trying to prove all of the theorems and solve all of the exercises, but I skipped a few theorems and exercises in the Field Axiom section (I 3.2 & I 3.3). The lack of continuity irks me, so I sometimes flip...
  7. M

    Checking Axioms for a*b on the Set Z

    Homework Statement For each of the following definitions for a * b and a given set, determine which of the axioms G0, G1, G2, G3 are satisfied by a * b. In which cases do we obtain a group? 1) a - b on the set Z 2)a + b - ab on the set R | {1} 3)ab on {2^n | n in Z} Homework...
  8. M

    Help Proof of simple theorems with addition and mulitplication axioms

    Thanks in advance. 1st day at calculus teacher wants proofs. They seem rudimentary but I've never done them and he doesn't help so I'm hoping someone here could please. These are the axioms: Addition: For a, b, and c taken from the real numbers A1: a+b is a real number also (closure) A2...
  9. T

    Proving the Vector Addition Axiom in Linear Algebra

    Homework Statement Prove that \vec0 + \vec{A} = \vec{A} The Attempt at a Solution \vec{A} + (-\vec{A}) = \vec0 \vec0 + \vec{A} = \vec{A} + (-\vec{A}) + \vec{A} = \vec{A} + \vec{A} + (- \vec{A}) \vec0+ \vec{A} = \vec{A} + \vec0 = \vec{A} I find this last line somewhat unconvincing...
  10. J

    Independence of Vector Space Axioms

    Homework Statement Determine whether the commutativity of (V,+) is independent from the remaining vector space axioms. Homework Equations N/A The Attempt at a Solution I am having a really hard time with this problem. Off the top of my head I could not think of any way to prove...
  11. T

    Proving the Associative Property for Polynomials in Linear Algebra

    Homework Statement Show that the axiom \vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C} holds for polynomials of the form a_0 + a_1 x + a_2 x^2 The Attempt at a Solution I'm pretty new to writing proofs for linear algebra so my first question is should I be treating the...
  12. L

    Mathematical Axioms of General Relativity

    What are the equations from which all of GR can be derived? Obviously one of the equations is Einstein's Field Equation: G^{\alpha\beta}=8\pi T^{\alpha\beta}. I would also guess that you would need the Euler-Lagrange Equations: -\frac{d}{d\sigma}(\frac{\partial L}{\partial...
  13. H

    Prove using three basic probability axioms

    Homework Statement Prove that P(A \cap B)≥1-P(\bar{A})-P(\bar{B}) for all A, B \subseteq Susing only these axioms: 1) 0 \leq P(A) \leq 1, for any event A \subseteq S 2) P(S) = 1 3) P(A \cup B) = P(A) + P(B) if and only if P(A \cap B) = 0Homework Equations None. The Attempt at a Solution My...
  14. T

    Is \pi a Homomorphism of Lie Algebras for \mathfrak{g} and \mathfrak{h}?

    Homework Statement Let \mathfrak{g} be a lie algebra over \mathbb{C} and \mathfrak{h} be an ideal of \mathfrak{g}. Show that the map \pi : \mathfrak{g} \to \mathfrak{g/h} defined by \pi (x) = x + \mathfrak{h} for all x\in\mathfrak{g} satisfies all the axioms of a homomorphism of lie...
  15. P

    Books on Axioms: ZF(C) & Why It's Complete

    Does anyone know of any good books on Axioms. Such as how was ZF(C) came up with and why it is that the general consensus is that it is complete.
  16. A

    Axioms of Probability: Deriving P(A∪B) = P(A) + P(B) - P(A∩B)

    I came across this question: http://imageshack.us/m/3/4510/axiomsq.png which I'm confused about. I know what the axioms of probability are but how would I use them to "derive" that result? I could illustrate why P(A∪B) = P(A) + P(B) - P(A∩B) on a Venn diagram but I have no idea how to use the...
  17. T

    Question regarding vector spaces and axioms

    Homework Statement I have quite a few problems that I believe I answered correctly, but here is one of them: 1. Rather than use the standard definitions of addition and scalar multiplication in R3, suppose these two operations are defined as follows: (x1, y1, z1) + (x2, y2, z2) = (x1+ x2...
  18. micromass

    Anybody else sick of the current math axioms?

    Anybody tired of those long, tedious proofs in ZFC? Anybody tired of those annoying counterexamples to beautiful results? Want to see a new kind of mathematics where everything works fine? Try Falso: http://estatis.coders.fm/falso/
  19. A

    What are the Fundamental Logical Axioms?

    Hi, I've been recently reading about logic. Is there a list of the exact logical axioms underlying all axiomatic systems, postulates and mathematics? Thanks...
  20. dextercioby

    What theories address the fundamental questions about quantum mechanics?

    I propose this set of axioms and ask you to be bring arguments for it and arguments against it. What other axioms would you choose instead of the ones I wrote below ? 1. STATE DESCRIPTION: All physical states of a quantum system are described mathematically by a set at most countable of...
  21. A

    What Are the Axioms of Algebra?

    Can someone please help me?
  22. M

    Proof using axioms for a field

    Hi, I am trying to work through Finite Dimensional Vector Spaces by Halmos, and I am having some difficulty with the first problem on page two (the specific problem is included below). The last class I took involving formal proofs was linear algebra about 8 years ago, and I am very rusty, but I...
  23. rpt

    What are the fundamental concepts and axioms of quantum mechanics?

    What are the axioms of quantum Physics? Does time/space considered fundamental quantities that considered to exist in nature when formulating quantum theory?
  24. R

    Proving 2+2=4 Using Field Axioms

    I've looked around but haven't found anyway to prove 2+2=4. I'm pretty sure you need to use field axioms, but I just haven't found it yet. Is there a way to do it? Like showing a+a=2a? Or a+b=c? Like 1+1=2. Something like that. Thanks!
  25. G

    Prove that F satisfies all field axioms by method of direct verification

    Homework Statement Consider the collection F of all real numbers of the form x+y√2, where x and y are rational numbers. Prove (by direct verification) that F satisfies all the field axioms (just like R) under the usual addition and multiplication. Homework Equations Field axioms...
  26. R

    Proving 0 = -0: Axioms & Solutions

    Homework Statement prove : 0 = -0 Homework Equations The Attempt at a Solution
  27. silvermane

    Comparison Proof via axioms, almost done need hints for finish and proof read

    1. The problem statement: Prove that 0\leq2x^2 - 3xy + 2y^2 2. These are the axioms we are permitted to use: 01) Exactly one of these hold: a<b, a=b, or b<a 02) If a<b, and b<c, then a<c 03) If a<b, then a+c < b+c for every c 04) If a<b and 0<c, then ac<bc. The Attempt at...
  28. silvermane

    Analysis Help; proofs via axioms

    Analysis Help; proofs via axioms :) 1. The problem statement: Prove that for any real numbers a, b, c, (a+b+c)^2\leq3*(a^2 +b^2+c^2) 2. These are the axioms we are permitted to use: 01) Exactly one of these hold: a<b, a=b, or b<a 02) If a<b, and b<c, then a<c 03) If a<b, then...
  29. D

    Explore Dependence of Axioms for Rings & Commutative Rings

    Homework Statement This is not an assignment question, just something that I am wondering about as an offshoot of an assignment question. In my course notes Rings are defined as having 3 axioms and commutative rings have 4.(outined below) I have just answered this question: Show that the...
  30. D

    How can you prove this using only the ring axioms?

    Homework Statement Using only the ring axioms, prove that in a general ring (R, +,X) aX (x-z) = (aXx)- (aXz) where all a,x,z are elements of R Homework Equations Group axiom 3: G3= There is an inverse for each element g^-1 *g =e Ring axiom 3: R3= Two distributive laws...
  31. J

    Proving a+b=b+a Using Ring Axioms

    Homework Statement Show that a+b = b+a follows from the other ring axioms. Homework Equations a + 0 = 0 + a = a (?) The Attempt at a Solution I know this is probably a simple algebraic manipulation, probably with the distributive law, but I don't know how to start! I'm sure I need...
  32. S

    Linear Algebra: Find all axioms that fail k(a,b)=(ak^2,bk^2).

    Homework Statement If we define V as the set of vectors in R^2 with vector addition defined as it normally is, but scalar multiplication defined to be k(a,b)=(k2a,k2b), then V is not a vector space. Find all axioms that fail (and explain why they fail). Homework Equations Axioms: 1. If u and...
  33. D

    Vector Space Axioms: Proving Axiom 1

    Since I can't copy and paste from maple into this message w/out losing formatting, I attached a pdf with all the work. I am having trouble proving axiom 1 of two general magic square matrices added together; plus, I am not sure if my set notation is entirely correct.
  34. S

    Axioms & Faith: What's the Difference?

    I had a question about axioms. Assuming I understand this correctly, axioms can neither be proven nor disproven; they are self-evident definitions that we have made to simplify math. So someone (with a strong religious motivation I might add) said that axioms are based on faith. You can't...
  35. S

    Can Mathematical Axioms Be Arbitrary?

    Hello everyone. I was wondering wether people think that mathematical conclusions can be provisional? I know about conjectures, but are they really part of maths? Finally, my definition of an axiom is: self evident and accepted Do you people think that there are better definitions than...
  36. S

    Proving Vector Space Axioms: (-1)u=-u

    Hi. please anyone help me with vector spaces and the way to prove the axioms. like proving that (-1)u=-u in a vector space.
  37. O

    Set theory: Axioms of Construction

    Homework Statement We're asked to prove that a few constructions of the sets a,b are themselves sets, stating which axioms we use to do so. a) a\b b)the function f:a->b c)the image of f Homework Equations The following standard definitions of axioms of construction...
  38. T

    Conventions for Natural Numbers: Debate & Axioms

    what is the convention you adhere to when it comes to natural numbers? for example there is a long standing debate about 0... should we define \mathbb N = \{0,1,2,...\} or instead \mathbb N = \{1,2,3,...\} and more about this, considering Peano's Axioms than we could choose \mathbb N...
  39. S

    Prove 1x = x from linear space axioms

    Homework Statement "Prove that Axiom 10[, the existence of identity in a linear space,] can be deduced from the other axioms." Homework Equations Now, I know that these axiom numberings are fairly arbitrary, but I'll put them in anyways for easy reference. I'm listing them word for word...
  40. I

    What Are Shankar's Inner Product Axioms in Quantum Mechanics?

    I was reading "Principles of Quantum Mechanics" - Shankar, and I'm having trouble understanding the inner product. Can someone help me or link me to a site that explains it? The axioms of the inner product are 1. \langle V|W\rangle = \langle W|V\rangle^* 2. \langle V|V\rangle \geq 0\ \...
  41. apeiron

    Is Mathematical Objectivity a Myth?

    Are the axioms of math subjective? If they are, then logically all the formal consequences that flow from them are also subjective. Thus there can be no objective mathematical facts. Someone said this: Someone then replied this: But then later made the contrasting statement: A...
  42. K

    Prove the following (using some basic axioms)

    Homework Statement Prove that if 0 < a < b, then a < \sqrt{ab} < \frac{a+b}{2} < b Homework Equations Axioms (Properties), courtesy of Wikipedia: Addition: P1: For all a, b, and c in F, a + (b + c) = (a + b) + c P2: There exists an element of F, called the additive identity...
  43. A

    Question about ordered field axioms

    If you have a > b and c \geq d, do you have a + c > b + d?
  44. R

    Axioms & Theorems: What's the Difference?

    Hi Can anyone help me define the axioms and theorems and what the differences are? I know axioms are suppose to be statements that are considered true based on logic (ex. x+y=y+x) but cannot be proven. Can someone explain why it can't be proven? Thanks
  45. E

    Exploring Perplex Numbers: R2 and the Field Axioms

    Homework Statement On R2, define the binary operators (x,y)+(u,v)=(x+u,y+v) (x,y)+(u,v)=(xu+yv,xv+yu) The set R2, along with these definitions of addition and multiplication, for the perplex numbers. (a) Show that \cdot: R2 \rightarrow R is associative...
  46. R

    Proof using the Axioms, Analysis

    1) For any x > 0 and 0 ≤ h < 1 we have (x + h)^2 ≤ x^2 + h(2x + 1). 2) For any x > 0 and p > 0 with x^2 < p there exists y > x with y^2 < p. Prove the following statements (only using the axioms for the real numbers). At each step say which axiom you use. The problems is that my professor...
  47. O

    Proving (-a)(-b)=ab Using Field Axioms

    hall, i need to prove by using the field axioms that: (-a)(-b)=ab, i think i know how to this but I'm very insecure with using those axioms cause i want to make sure I'm not using my intuition. i tried something like: (-a)(-b)=(-1)(a)(-1)b=(-1)(-1)(a)(b)=ab and i guess it's wrong...
  48. J

    Determine if the space is a subspace testing both closure axioms

    Homework Statement determine if the space is a subspace testing both closure axioms. in R^2 the set of vectors (a,b) where ab=0 Homework Equations The Attempt at a Solution i just used the sum and product which are the closure axioms. But at the end how do you tell if the...
  49. F

    Squaring a vector and subspace axioms

    If a problem I'm doing asks to find V2 where V is a vector is it simply the dot product of the vector, or the cross product? The question: Which of the following sets of vectors v = {v1,...,vn} in Rn are subspaces of Rn (n>=3) iii) All v such that V2=V12 He proved it by saying...
  50. D

    Why do we use the term 'axioms' for vector spaces instead of 'definitions'?

    Why are they called "axioms"? Shouldn't they be called "definitions"?
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