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. starstruck_

    I The Nuances of Truth in Axioms and Premises

    Hey everyone, I’m taking my first discrete math course this term and am kind of struggling with determining the difference between different terminology. As the title says, it’s specifically with premises and axioms. My professor’s notes begin with an introduction to premises as one of the two...
  2. N

    I Can interpretation-dependent facts be derived from the axioms?

    In the thread https://www.physicsforums.com/threads/post-selection-pre-existing-correlations-or-action-at-a-distance.1049354/, @PeterDonis claims that a certain mathematical derivation from the basic axioms of QM is an interpretation-dependent proposition. I'm referring to post #54 here and this...
  3. D

    I Prove $$T_{p}M$$ is a vector space with the axioms

    Suppose M is a manifold and $$T_{p}M$$ is the tangent space at a point $$p \in M$$. How do i prove that it is indeed a vector space using the axioms: Suppose that u,v, w $$\in V$$. where u,v, w are vectors and $$\V$$ is a vector space $$u + v \in V \tag{Closure under addition}$$ $$u + v = v +...
  4. chwala

    Understanding of the Metric Space axioms - (axiom 2 only)

    Am refreshing on Metric spaces been a while... Consider the axioms below; 1. ##d(x,y)≥0## ∀ ##x, y ∈ X## - distance between two points 2. ## d(x,y) =0## iff ##x=y##, ∀ ##x,y ∈ X## 3.##d(x,y)=d(y,x)## ∀##x, y ∈ X## - symmetry 3. ##d(x,y)≤d(x,z)+d(z,y)## ∀##x, y,z ∈ X## - triangle inequality...
  5. plum356

    Writing some ZF axioms with FOL symbols

    The axioms: My work: Extension:$$\forall x\forall y,\,(x=y)\iff(\forall z,\,(z\in x\iff z\in y))$$ Empty Set:$$\exists x|\forall y,\,\neg(y\in x)$$ Pair Set:$$\forall c\forall d,\,\exists e|(c\in e)\wedge(d\in e)\wedge[\forall f,\,\neg((f=c)\vee(f=d))\implies\neg(f\in e)]$$ If you consider any...
  6. BohmianRealist

    A Bootstrapping quantum Yang-Mills with concrete axioms

    From: https://en.wikipedia.org/wiki/Axiomatic_quantum_field_theory But, that seems like a fairly abstract place to begin the kind of QFT construction that was asked of us by Witten in 2012: At the bottom of that page on axiomatic QFT are the "Euclidean CFT axioms": Are there any examples of...
  7. S

    Thermodynamic Axioms: Establishing Temperature, Internal Energy & Entropy

    Is the purpose of the 0th, 1st & 2nd Laws of Thermodynamics simply to legitimate the thermodynamic properties of Temperature, Internal Energy & Entropy, respectively? It seems that all these laws really do is establish that these properties are valid thermodynamic state properties and the...
  8. dextercioby

    I One more talk about the independence of Einstein's SR axioms

    Sorry if this is discussed here previously, but I just stumbled upon an article from 1911 which I would like to bring forth to you. Preamble: it is generally thought that Einstein's (refined) two axioms of SR (1. The laws of physics are invariant upon shifting from one IRF to another. 2. The...
  9. M

    MHB Subsets of $\mathbb{R}^2$ Satisfying S2 and S3 but Not S1: Empty Set

    Hey! :giggle: The three axioms for a subspace are: S1. The set must be not-empty. S2. The sum of two elements of the set must be contained in the set. S3. The scalar product of each element of the set must be again in the set. I have shown that: - $\displaystyle{X_1=\left...
  10. penroseandpaper

    I Subgroup axioms for a symmetric group

    Hi, The textbook asks me to use subgroup axioms to prove why a set of permutations that interchange two specific symbols in S4 is or isn't a subgroup of the symmetric group, and the same for a set of permutations that fix two elements. My guess is that the set of permutations that interchange...
  11. J

    A An additional constraint to the ZFC axioms?

    The ZFC axioms are statements combining "atomic formulas" such as "p ∈ A" and "A = B", using AND, OR, imply, NOT, for all and exists. But (it seems to me, at least) there is the implicit assumption that the "atomic formulas", "p ∈ A" and "A = B", are considered to be propositions, i.e. they are...
  12. M

    MHB Subset that satisfies all but one axioms of subspaces

    Hey! :o I want to find subsets $S$ of $\mathbb{R}^2$ such that $S$ satisfies all but one axioms of subspaces. A subset that doesn't satisfy the first axiom: We have to find a subset that doesn't contain the zero vector. Is this for example $\left \{\begin{pmatrix}x \\ y\end{pmatrix} ...
  13. S

    MHB Proving Primitive Symbols with Axioms

    Given : A) primitive symbols : (1, *) and B) The axioms: 1) \forall x\forall y[x*=y*\Longrightarrow x=y] 2) \forall x[x*\neq 1] 3) [P(1)\wedge\forall x(P(x)\Longrightarrow P(x*))]\Longrightarrow\forall xP(x) Then prove: \forall x[x=1\vee \exists y(y*=x)]
  14. M

    Field axioms - is there an axiom for multiplication with zero?

    Please refer to the screenshot below. Every step is justified with an axiom. Please see the link to the origal document at the bottom. I am trying to understand why the proof was not stopped at the encircled step. 1. Is there no axiom that says ## x \cdot 0 = 0 ## ? 2. Isn't the sixth...
  15. A

    MHB Treatment of axioms of formal axiomatic theory

    What is the proper treatment of results about a formal axiomatized theory which are obtained from outside the theory itself? For example, there are 9 results dealing with the "≤" relation for Robinson Arithmetic, some of which are established by using induction, which is not "native" to Q...
  16. ali PMPAINT

    B Proving congruent with Euclidean axioms

    So, given one, you can prove the others, but I don't know how to prove one with using the five axioms.
  17. A

    MHB Question about identity axioms

    I'm going through Peter Smith's book on Godel's Theorems. He mentions a simple formal theory ("Baby Arithmetic") whose logic needs to prove every instance of 'tau = tau'. Does every 'standard deductive apparatus' include the common identity axioms (e.g. 'x = x')?. The axioms of "Baby...
  18. M

    MHB The axioms of a vector space are satisfied

    Hey! :o We consider the $\mathbb{F}_2$-vector space $(2^M, +, \cap)$, where $M$ is non-empty set and $+ : 2^M\times 2^M \rightarrow 2^M: (X,Y)\mapsto (X\cup Y)\setminus (X\cap Y)$. I want to show that $(2^M, +, \cap )$ for $\mathbb{K}=\{\emptyset , M\}$ satisfies the axioms of a vector space...
  19. N

    Axioms of Probability: Cell Phone Factory

    Homework Statement Give an factory of cell phones there is a .5 rejections, .2 repaired, and .2 acceptable. Does this follow the axioms of probability. Homework Equations Sample space = 1; Probaby: 0 -1 P(AnB)=P(A)+P(B) The Attempt at a Solution Technically this does follow the axioms, there...
  20. M

    MHB Which of the group axioms are satisfied?

    Hey! :o I want to check the following sets with the corresponding relations if they satisfy the axioms of groups. $M=\mathbb{R}\cup \{\infty\}$ with the relation $\min:M\times M\rightarrow M$. It holds that $\min (a, \infty)=\min (\infty, a)=a$ for all $a\in M$. $M=n\mathbb{Z}=\{n\cdot...
  21. nomadreid

    I Repeatability of necessity: number restrictions?

    The necessity quantifier (aka Provability quantifier, or ~◊~, or Belief, or... instead of the usual square I will be lazy and call it "N") is often allowed to be repeated as many (finite) times as one wishes, so NNNNNNψ is OK. Is it possible to somehow include into the axioms some restriction on...
  22. Jarvis323

    A Choices of Axiomatic and Number Systems / Sets and Alternatives

    I know that the number systems we use are typically constructed from axiomatic set theory, and overall our choices along the way seam to have been largely informed by practical consideration (e.g. to resolve ambiguities, or do away with limitations). Today I randomly started to think deeper...
  23. R

    MHB Axioms of Sinus Definition

    My class is difficult to teach, but I have a question that I think that I share the forum and if you give nice ideas it can be helpful. This is my last question of axioms and so on because I don't want to be as a mathematic cranck. So, this is my last question that deal with it. What I need to...
  24. S

    MHB Proving Identities Using Axioms of Equality

    Given the following axioms: For all A,B,C...we have: 1) A=A 2) A=B <=> B=A 3) A=B & B=C => A=C 4) A=B => A+C= B+C 5) A=B=> AC =BC ( NOTE :Instead of writing A.C or B.C e.t.c we write AB.BC e.t.c) 6) A+B= B+A..........AB=BA 7) A+(B+C) = (A+B)+C............A(BC)=(AB)C 10)...
  25. R

    MHB Integration Basics: Axioms & Answers

    One of my students ask me: "Which axioms are the basic of the integration?" What I should answer him? Any ideas?
  26. A

    I Proof that 1 is an odd number using Peano Axioms of naturals

    So I was just writing a proof that every natural number is either even or odd. I went in two directions and both require that 1 is odd, in fact I think that 1 must always be odd for every such proof as the nature of naturals is inductive from 1. I am using the version where 1 is the smallest...
  27. K

    I Exploring the Origins of Topology Axioms

    Is there a way we can see why the axioms defining a topology/ topological space are the way they are?
  28. S

    Prove a statement using Peano's Axioms

    Homework Statement Let, m, n be natural numbers and S(n) the succesor of n. If S(n)*m = nm + m Prove that m*S(n) = nm + m Homework Equations The Attempt at a Solution
  29. N

    A Building on QP from 5 reasonable axioms

    Lucien Hardy's Quantum Theory From Five Reasonable Axioms has deepened my understanding of QP foundations, and motivated me to write a paper. The essence of my paper is that "connectedness" of state space (or the acting Lie group), need not be assumed, but can be deduced. Before linking to the...
  30. Math Amateur

    MHB Axioms of Set Theory .... and the Union of Two Sets ....

    I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ... I am currently focused on Chapter 1: Sets and, in particular on Section 3: The Axioms where Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ...
  31. U

    Determining a group, by checking the group axioms

    Homework Statement For the following sets, with the given binary operation, determine whether or not it forms a group, by checking the group axioms. Homework Equations (R,◦), where x◦y=2xy+1 (R*,◦), where x◦y=πxy and R* = R - {0} The Attempt at a Solution For question 1, I found a G2...
  32. parshyaa

    I Question about Euclid's parallel postulate (5th postulate).

    Why can't we prove euclids fifth postulate What's wrong in this proof: why can't we prove that there is only one line which passes through a single point which is parallel to a line. If we can prove that two lines are parallel by proving that the alternate angles of a transverse passing...
  33. L

    I Conjectures, Hypotheses, Axioms

    Intuitive representation of the differences amongst Conjectures, Hypotheses and Axioms. @fresh_42 @FactChecker @WWGD
  34. T

    I How can -a be equal to (-1)*a?

    I will say that this question is coming from a lack of explanation in a classroom, however this particular proof is not homework and is just explanation over a proof that was discussed briefly in class, so I didn't think it belong in the homework section. I'm also not certain it belongs in the...
  35. Math Amateur

    MHB ZFC .... Axioms of Foundation .... and Infinity ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume 1: Foundations and Elementary Real Analysis" ... ... I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ... I need help with an aspect of the proof of...
  36. Math Amateur

    I ZFC .... Axioms of Foundation .... and Infinity ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume 1: Foundations and Elementary Real Analysis" ... ... I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ... I need help with an aspect of the proof of...
  37. S

    I What is an Axiom? Definition & Examples

    Hello! I was wondering how does a mathematical statement come to be an axiom? I understand that an axiom can't be proven using other mathematical statements. But how does one know that a statement can or can not be proven? For example, why isn't Riemann Hypothesis considered an axiom? I also...
  38. Y

    MHB Proofs Using Axioms Only

    Hello all, I am looking for simple theorems that can be proved by using Hilbert's axioms of Geometry only. For example, such a theorem can be "two lines intersect in a single point". I am looking for more examples that can be proved (with a short proof) using these axioms. Can you think of such...
  39. M

    B Are ZF axioms necessary?

    Obviously information exists. From the concept of information quanta we can create physical models of mathematical concepts. Are ZF axioms redundant?
  40. L

    I How to define addition and multiplication using Peano's axioms?

    How can one define addition using peanos axioms? Number, successor, zero are terms which we presume to know the meaning of. We then use five propositions: 1. 0 is a number. 2. Every number has a successor. 3. 0 is not the successor of any number. 4. Any proeprty common to zero and its...
  41. Math Amateur

    MHB Axioms of Set Theory: Separation Axiom and Garling Theorem 1.2.2 .... ....

    I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...
  42. M

    I Cancellation law multiplication natural numbers

    Hello everyone. I wanted to prove the following theorem, using the axioms of Peano. Let ##a,b,c \in \mathbb{N}##. If ##ac = bc##, then ##a = b##. I thought, this was a pretty straightforward proof, but I think I might be doing something wrong. Proof: Let ##G := \{c \in \mathbb{N}|## if ##a,b...
  43. M

    A If the axiom of induction were extended to include imaginary numbers....

    If the axiom of induction was extended to include imaginary numbers, what effect would this have? The axiom of induction currently only applies to integers. If this axiom and/or the well ordering principle was extended to include imaginary numbers, would this cause any currently true statements...
  44. beep300

    I General topology: Countability and separation axioms

    I need some help understanding the countability and separation axioms in general topology, and how they give rise to first-countable and second-countable spaces, T1 spaces, Hausdorff spaces, etc. I more or less get the formal definition, but I can't quite grasp the intuition behind them. Any...
  45. D

    MHB Creating theorems from the given axioms

    Given the following axioms, create 3 theorems. Axiom 1: Each game is played by two distinct teams. Axiom 2: There are at least four teams. Axiom 3: At least six games are played. Axiom 4: Each team plays at most four games. I have already proven each axioms independence. These...
  46. NoName3

    MHB Using the axioms of ordered field

    I only have the axioms of ordered plus the basic laws of equality and inequalities that follow from it. 1. Prove that $1 < 2$. 2. Prove that $0 < \frac{1}{2} < 1$ 3. Prove that if $a, b \in \mathbb{R}$ and $a < b$ then $a < \frac{a+b}{2} < b$ 1. Assume the opposite - that's, suppose that...
  47. Zafa Pi

    I Are the 2nd and 3rd axioms of QM incompatible?

    Nielsen & Chuang list three axioms for QM. I paraphrase them as follows: 1. States are unit vectors. 2. The evolution of a state is unitary and given by the Schrodinger equation. 3. The measurement of a state yields a value from a probability distribution. The state just before the...
  48. L

    "Feeling" the relation of math to the real world

    I am not a mathematician but, as such, I think I have a pretty good background in mathematics. I have a good understanding and experience with calculus, differential equations, linear algebra, and probability theory. I also have interest in abstract algebra concepts, though I wouldn't say I am...
  49. Demystifier

    I A confusion about axioms and models

    Suppose that I have a set of axioms in first-order logic. And suppose that I have several inequivalent models for this set of axioms. And suppose that I want to choose one specific model. To choose it, I need to make some additional claims which specify my model uniquely. My question is the...
  50. C

    Question about the axioms of set theory

    Homework Statement For each structure, draw a directed graph representing the membership relation. Then determine which of the following axioms is satisfied by the structure: Extensionality, Foundation, Pairing, Union U= {a,b} a in b , and b in a The Attempt at a Solution The directed...
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