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. N

    MHB Justifying Independence of Axioms 1-4

    I need help with the following situation--- I need to justify the independence of all four axioms using computation and/or narrative. This is what I have so far, but would appreciate any ideas and help you may have to offer. Axiom 1: Each game is played by two distinct teams. Axiom 2: There...
  2. J

    Looking for book on ZFC axiomatic set theory (I think)

    I am looking for a book that starts at the standard ZFC axioms and progresses to the point where some recognisable non-trivial mathematical statement is proved. By recognisable I mean something that you may encounter in school/early university level and is not purely set-theoretical (e.g...
  3. G

    MHB Checking Field Axioms for $G = F \times F$

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  4. naima

    Test functions in wightman axioms

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  5. F

    Exploring the Relationship Between Axiom Count and Theory Validity

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  6. E

    Solving Vectors & Axioms Homework

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  7. vrble

    Derivation of the Area Formula for Triangles Using Axioms

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  8. I

    Field axioms, subspaces

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  9. C

    MHB Proving Independence of Axioms in Game Theory: A Case Study

    I have four axioms and I am stuck trying to prove the independence of these axioms. Axiom 1: Each game is played by two distinct teams. Axiom 2: There are at least four teams. Axiom 3: Exactly six games are played. Axiom 4: Each distinct team played once against the same team. I've justified...
  10. W

    My simple proof of x^0=1 part 2 (axioms)

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  11. Aristotle

    Need help with proof of Vector Space (Ten Axioms)

    Let S={x ∈ R; -π/2 < x < π/2 } and let V be the subset of R2 given by V=S^2={(x,y); -π/2 < x < π/2}, with vector addition ( (+) ). For each (for every) u ∈ V, For each (for every) v ∈ V with u=(x1 , y1) and v=(x2,y2) u+v = (arctan (tan(x1)+tan(x2)), arctan (tan(y1)+tan(y2)) )Note: The...
  12. D

    Why don't these open set axioms specify that the empty set is open?

    In all the topology textbooks I used in school, the open set axoims specified 4 conditions on a set S: (i) S is open (ii) empty set is open (iii) arbitrary union of open sets is open (iv) finite intersection of open sets is openI noticed on proofwiki, that (ii) is omitted. I was curious if...
  13. G

    Examples in lin.alg. where closure/addition axioms don't hold

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  14. Dethrone

    MHB Exploring $\equiv$, =>, and <=> Symbols

    First day of school and I don't have much work. So I got bored and read ahead a bit. From a textbook my prof wrote: Can someone explain why they used the "$\equiv$"? I think it means "equivalent", but I'm not sure, but when are times you want to use that symbol rather than "equals"? What's the...
  15. S

    Explanation of Axioms, First Principles, and Assumptions

    College sophomore here and just wanted to ask for an explanation of terms that I have come across in my math and philosophy courses thus far, but without direct teaching of them. I've seen these terms used in forum discussions and mentioned off-handedly by profs., but don't have a specific...
  16. A

    Proving P(B)>=P(A) using Kolmogrov's axioms

    Homework Statement Suppose A is a subset of B. Using the three axioms to establish P(B)≥P(A) Homework Equations Axioms... 1. For any event A, P(A)\geq0 2.P(S)=1 3.If events A1,A2... are mutually exclusive, then P(\cupAi)=\sumP(Ai) The Attempt at a Solution From the venn...
  17. snoopies622

    Necessary axioms to derive solution to QHO problem

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  18. sheldonrocks97

    Prove that C[a;b] satisfies 8 axioms of vector space

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  19. A

    Lingusitics Are there axioms in natural language?

    In math, we have axioms that we assume to be true. We don't have proofs for it. Similarly english or any other natural language attempts to describe the world using words, alphabets etc. So there must be some axioms, right? Everything cannot be described / defined. But obviously, we can...
  20. E

    What are the Zermelo-Fraenkel Axioms and their meanings?

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  21. S

    MHB Geometry Book: 2 Axioms & Their Implications

    In a Goemetry book i read the following two axioms. 1) There exist at least two different points on each straight line 2) There is exactly one line on two different points But the 2nd doesn't imply the 1st axiom??
  22. N

    How do axioms for Euclidean geometry exclude non-trivial topology?

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  23. M

    Why is Axiom of Induction Needed as an Axiom? (Peano's Axioms)

    I've already asked somebody through email this question, so I'll copy and paste part of my email: Basically, I'm wondering why doesn't it fall from the other axioms, and if it does in fact not fall from the other axioms (which it apparently doesn't), why the axioms can't be slightly modified...
  24. N

    MHB Questions on Basic Axioms for Calculating Probabilities

    Please refer to the attached image. for problem 1a) i can see how this makes intuitive sense, however the hint confuses me. When we are told that $A_n \subset A_{n+1}$ why would the hint say to attempt to express $A$ as as a union of countably many disjoint sets, when it is defined not to be...
  25. B

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  26. S

    MHB Proving 0<1 with Axioms: A+B=B+A, A.B=B.A, and More!

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  27. A

    How do we know axioms are sufficient?

    In math, we define new concepts and have certain axioms regarding the concept that are believe to be true. Now how do we know that the given axioms are enough to give a proof to any problem? I hope this makes sense.
  28. A

    Probability Axioms: Explaining Their Definition

    Axioms are: - 1) P(E) >= 0 2) P(S) = 1 3) P(E1 U E2 U ...) = P(E1) + P(E2) + ... if all are mutually exclusive Why are the axioms defined in such a way? Why not this simple axiom: - Probability of an event is number of favorable outcomes divided by total number of outcomes?
  29. paulmdrdo1

    MHB Axioms for the real numbers.

    in this problem we drop the use of parentheses when this step is justified by associative axioms. thus we write $\displaystyle x^2+2x+3\,\,instead\,\,of\,\,\left(x^2+2x\right)+3\,or\,x^2+\left(2x+3\right)$. tell what axioms justify the statement: 1. $\displaystyle...
  30. R136a1

    Difficulty checking group axioms

    Let ##G## be a set equipped with a binary associative operation ##\cdot##. In both of the following situations, we have a group: 1) ##G## is not empty, and for all ##a,b\in G##, there exists an ##x,y\in G## such that ##bx=a## and ##yb=a##. 2) There exists a special element ##e\in G##...
  31. S

    Is it possible to calculate in physics with different sets of axioms?

    I was just wondering, is it possible? It's regarding a debate on whether mathematics is an invention or discovery.
  32. X

    If we had enough axioms about reality could mathematics replace physic

    Math is able to prove and disprove things while science can only disprove. The reason we can't prove much about the actual world is because unlike in mathematics we don't have enough starting axioms. If the math world isn't real and is just an object of the mind it would make sense that us...
  33. TrickyDicky

    What is the most fundamental axiom of General Relativity?

    I would like to get some foundational concepts straight about GR as it is currently understood (I guess it is not seen exactly the same it was almost a century ago even if the basic concepts remain, this I would like to elucidate here too). For instance, I understand that a basic axiom of GR as...
  34. C

    On the axioms of vector space

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  35. D

    Can Sets be Defined from Peano Axioms Alone?

    Is it possible to define sets from just the peano axioms? Usually when people use the peano axioms as the basis of their math they just assume the existence of sets but without axioms regarding sets we technically shouldn't just say they exist. Oh, also there are two versions of the...
  36. V

    What actually are Newton's axioms of classical mechanics

    Just like the Euler-Lagrange differential equation $$ \frac{\partial{\mathcal{L}}}{\partial{q}} = \frac{d}{dt}\frac{\partial{\mathcal{L}}}{\partial{\dot{q}}} $$ the hamiltonian equations $$ \frac{\partial{H}} {\partial{q}} = -\dot{p} $$ $$ \frac{\partial{H}} {\partial{p}} = \dot{q} $$ and the...
  37. phosgene

    Another proof using the axioms of probability

    Homework Statement If A and B are events, use the axioms of probability to show that: if B \subset A, then P(B) \leq P(A) Homework Equations Axiom 1: P(n) \geq 0 Axiom 2: P(S)=1 Axiom 3: If A1,A2,... are disjoint sets, then P(\bigcup _{i} A_{i}) = \sum_{i} P(A_{i}) The Attempt at a...
  38. phosgene

    Proving identities using the axioms of probability

    Homework Statement If A and B are events, use the axioms of probability to show: a) If A \subset B, then P(B \cap A^{C}) = P(B) - P(A) b) P(A \cup B) = P(A) + P(B) - P(A \cap B) Homework Equations Axiom 1: P(x)\geq 0 Axiom 2: P(S) = 1, where S is the state space. Axiom 3: If...
  39. A

    Peano axioms for natural numbers - prove 0.5 ∉ N

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  40. P

    Help with vector spaces axioms

    Homework Statement for the 2x2 matrix [a 12;12 b] is it a vector spaceHomework Equations 1. If u and v are objects in V, then u+v is in V 2. u+v = v + u 3. u+(v+w) = (u+v)+w 4. There is an object 0 in V, called a zero vector for V, such that 0+u = u+0 = u for all u in V 5. For each u in V...
  41. M

    Classical definition of probability & kolmogorovs axioms

    I've seen in some probability theory books that the classical definition of probability is a probability measure, it seems fairly trivial but what is the proof for this? Wikipedia gives a very brief one using cardinality of sets. Is there any other way?
  42. Logic Cloud

    Classical version of QM axioms

    Hello, I am looking for more information on the axiomatic treatment of physics. I have found some articles concerning the axioms of quantum mechanics, i.e. the Dirac-Von Neumann axioms. However, I am having a hard time finding anything on the classical version of these axioms. In these...
  43. N

    Using field axioms to prove a set is not a field

    Homework Statement Let F = {a + b\sqrt[3]{2}:a,b\inQ}. Using the fact that \sqrt[3]{2} is irrational, show that F is not a field. [Hint: What is the inverse of \sqrt[3]{2} under multiplication?] Homework Equations For a field, For all c \in F, there exists c-1 \in F s.t. c*c-1 =1...
  44. W

    Prove 2ab<= a^2+b^2 using order axioms

    Homework Statement prove 2ab<= a^2+b^2 using order axioms Of course use of other basic axioms for real numbers are also okay. Homework Equations axioms for set of real numbers. The Attempt at a Solution The easy way to do this would be just subtract 2ab from both sides, factor...
  45. H

    Showing Order Fields Axioms Hold: a<b for 0<a,b in F

    Homework Statement Using only axioms o1-o4 and the result in part (i), show that for any a,b\inF, with 0<a and 0<b, that if a^{2}<^{2}b, then a<b. Homework Equations o1) For any a,b\inF, precisely one of the three following holds: a<b, b<a, a=b o2) If a,b,c\inF, and if a<b and b<c, then...
  46. J

    Where do the vector space axioms come from?

    Every resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?
  47. 1

    The other day our lecturer was going through field axioms, rules of

    The other day our lecturer was going through field axioms, rules of numbers, I guess what you'd call very elementary number theory. In particular, he was explaining what you can and can't do with ∞, and he mentioned that x/∞=0. I guess I'd just like some clarification on this. Should this not...
  48. O

    MHB On the Implication of Addition Axioms

    Hello everyone! I was trying to prove the propositions that follow the addition axioms as a revision, I got a different proof for the following proposition: If $x+y=x+z$ then $y=z$ My proof was the following: $x+y=x+z$, $(-x)+x+y=(-x)+x+z$, $0+y=0+z$, $y=z$ Rudin however, in his book...
  49. T

    Linear Algebra Proofs: Solving for Scalar and Vector Using Axioms

    Hey all, So I'm just starting a course in linear algebra, but I don't have much experience with proofs. This problem has been giving me some difficulty. So we have a scalar "a" and vector x. V is a linear space, and x is contained in V. I have to show that if ax=0, where 0 is the zero...
  50. C

    Limit of Theorems with Countable Axioms: Exploring Mathematical Possibilities

    If I have an \aleph_0 number of axioms, does that put a limit on the number of theorems I can have about the real numbers. The number of theorems that we could have is countable. I was just wondering what we might be able to say about how much we could know about math.
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