What is Axiom: Definition and 147 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. E

    Axiom of Choice to prove two propositions.

    Hi everyone, we recently covered some implications of the AC and are now to prove the followings statements with the help of the AC or one of its equivalent: (1) Every uncountable set has a subset of cardinality \aleph_1 (the least initial ordinal not less or equal than \aleph_0, the latter...
  2. L

    Proof of Axiom of Choice equivalent.

    I'm trying to prove that the axiom of choice is equivalent to the following statement: For any set X and any function f:X\to X, there exists a function g:X\to X such that f\circ g\circ f=f. I was able to prove that the AoC implies this, but I'm having a harder time going the other...
  3. S

    Proving an Axiom: Stuck on Field's Commutativity

    Homework Statement I was asked to prove the axioms of a field. so, if we look at the first one: commutativity: a+b=b+a and a*b=b*a where a and b belong in the set of the field Homework Equations The Attempt at a Solution it's tempting to just substitute values but i know this...
  4. M

    Logic axiom of simplification.

    Hello all I cannot find a simple explanation of the meaning of this axiom, probably because it is considered so obvioius that it needs no explanation. Can anyone explain in words. {a}\rightarrow{({b}\rightarrow{a})} Thanks. Matheinste.
  5. B

    Understanding Completeness Axiom: Real Numbers & Subsets

    I'm having a little touble understanding application of the completeness axiom to certain subsets of real numbers. In a problem in a book (Fundamentals of Real Analysis by Haggarty), it asks you to show that the set S={a + b*sqrt(2) : a,b are rational} is not complete. As a hint, it tells you to...
  6. U

    Can you prove that 1+1=2 or is it an axiom?

    Can you prove that 1+1=2 or is it an axiom?
  7. C

    Is Our Perception of Causality Limited by Time and Perception?

    Do you agree with this? http://en.wikipedia.org/wiki/Axiom_of_Causality "The Axiom of Causality is the proposition that everything in the universe has a cause and is thus an effect of that cause. This means that if a given event occurs, then this is the result of a previous, related event...
  8. J

    Proving Surjective Functions with Finite Y: No Axiom of Choice Required

    Homework Statement Prove, without using the Axiom of Choice: if f: X->Y is surjective and Y is finite, there exists a 'section', a function s:Y->X such that f(s(y))=y for all y in Y Hint: perform induction over the cardinality of Y The Attempt at a Solution Induction over the...
  9. G

    Analyzing if 'x<y ⇒ x^2<y^2': What Axiom to Use?

    i was wondering if I'm using the right approach for this the question reads is the following statement true for all x and y : 'If x<y then x^2<y^2' then it follows by asking about 'if x^2<y^2' i am currently using case analysis to do this by considering whether x and y are positive or...
  10. F

    Induction Axiom and the Fallacy of All-Encompassing Conclusions

    Homework Statement The inductive hypothesis P(n): For any counting number n in N, and set of billiard balls with n members, all the balls have the same color. Pf) Consider any set A of n+1 balls, and the subsets B=(first n balls), C=(last n balls) The inductive hypothesis applies to both B...
  11. J

    Is axiom of choice applicationless?

    Does there exist a theorem, whose proof relies on the axiom of choice, and which has practical or concrete applications? I'll consider for example PDE and number theory problems practical or concrete, because they can be intuitively connected with the physical world. On the other hand, for...
  12. D

    Axiom of Choice: Unveiling the Mystery Behind Unsolvable Problems

    Can someone give me a list of problems which at first sight require the axiom of choice, but do not?
  13. W

    Dedekind-Cantor Axiom: What's Needed & Why?

    I know that the Dedekind-Cantor axiom establishes an isomorphism between the points of any given (extended) Euclidean line. But why is the axiom needed anyway? Can't we define two binary operations on collinear points in Euclidean geometry such that the points of the line taken together with...
  14. C

    ZF Axiom of Infinity: Natural Numbers Exist

    In ZF, the axiom of infinity says that the set of natural numbers exists. I was wondering if there was a (finitist?) weakening of ZF that included the axiom "the class of natural numbers exists".
  15. M

    Set Theory Regulatory Axiom and Ranks

    Homework Statement Assume that D is a transitive set. Let B be a set with the property that for any a in D, a is a subset of B implies a is an element of B. Show that D is a subset of B. The Attempt at a Solution My first step is to show that the empty set must be an element of D...
  16. M

    Do you believe in the Axiom of Choice?

    Feel free to give your reasons. I voted yes, because too many useful theorems are thrown out the window if Axiom of Choice is rejected. I believe that these useful theorems outweigh the surprising (strange?) results that also arise from AC (e.g. every set can be well-ordered). Also, if AC...
  17. K

    What's your opinions on the Axiom of Choice?

    I, by the first time, came across with the Axiom of Choice today, found it beautiful, of course. And I'm interested in seeing, ladies and gentlemen here, as mathematicians, what are your attitude towards that axiom, I mean, the stronger one (i.e. the infinite axiom of choice). For example, do...
  18. J

    Change of Axiom of Probability

    The reference book I have used stating that: Axiom 1 stating that 0<=P(E)<=1 Axiom 2 stating that P(S)=1 Axiom 3, the probability of union of mutually exclusive events is equal to the summation probability of of each of the events. And the author says that, hopefully, the reader will agree...
  19. J

    Axiom of Probability: Question on p(<sample space>)=1

    I have a question regarding one of the axiom for probability, which is p(<sample space>)=1. I do not understand why p(<sample space>)=1 is an axiom instead of theorem, since I can prove it with the following argument: Since sample space has been defined as the set of all possible outcomes...
  20. quasar987

    Axiom of Choice: Finite Character & Maximal Sets

    There is an axiom/lemma from Teichmilles & Tukey that is equivalent to the axiom of choice. It reads, Every family of sets F that is of finite character (http://en.wikipedia.org/wiki/Finite_character) possesses a maximal element. I just want to confirm that here, "maximal set" means a set...
  21. M

    Weak/strong group identity axiom

    Hello all. While going back to group theory basics to make sure i understand rather than just know the fundamentals i came across for the first time ( having read many books ) the weak versus strong versions of the identity axiom. The strong version says that a group must have a unique...
  22. L

    Wavefunction collapse: is that really an axiom

    Can the wavefunction collapse not be derived or is it really an axiom? How can the answer to this question (yes or no) be proven? If it is an axiom, is it the best formulation, is it not a dangerous wording? Let's enjoy this endless discussion !
  23. S

    Why Do We Need the Axiom of Choice?

    the axiom of choice. Help! well i am having trouble understanding why we need the axiom of choice. So could anybody post here some problems, and their solutions, that include the axiom of choice, and explain how it works, so how it is used in solving problems, and explain why we actually need...
  24. D

    The Axiom of the Power Set

    Hey everyone, I am currently trying to learn a bit of set theory from Halmos' book "Naive Set Theory" since I have recently been concerned with the general notion of existence in various fields of mathematics. Now, I am reading the "axiom of the power set" and I do find it a little...
  25. wolram

    Geometry of Time, Axiom of Choice, and Neuro-Biological Quantum Zeno Effect

    Is ther a mass loss at the time of death? Can brain states be frozen? [9] arXiv:0704.1054 [ps, pdf, other] : Title: Geometry of Time, Axiom of Choice and Neuro-Biological Quantum Zeno Effect Authors: Moninder Singh Modgil Comments: 13 pages Role of axiom of choice in quantum...
  26. MathematicalPhysicist

    Some questions on axiom of choice and zorn's lemma.

    1)axiom of choice: prove that for every set X and for every f:X->X there exists g:X->X such that fogof=f. 2)zorn's lemma: let R be a partial order on X (X a set), prove that there exists a linear order S on X such that R is a subset of S. well for the second question i used zorn's lemma to...
  27. T

    Axiom of choice with single-point sets

    axiom of choice do you need to invoke the axiom of choice to choose a point from a collection of sets if the sets are single-point sets? for example, suppose f:A->B is injective. to create a left inverse g:f(A)->A, we need to "choose" a point from the preimage of b for all b in f(A) and...
  28. MathematicalPhysicist

    Inquiry about axiom of choice.

    i have this question: if: X=U X_i for every i in I, where X_i's are non empty and are disjoint, then |X|>=|I|. obvously there's the one to one function from I to X, which is g(i)=X_i, but my question is according to my text i need to use here the axiom of choice, i don't think i used here the...
  29. S

    Zorn's lemma without the axiom of choice

    I'm wondering if there is a version of Zorn's lemma that applies to collections that are "small" in a sense I'll describe below, and which true independent of the axiom of choice. Specifically, say I have a collection of sets such that each set in it is countable, but the collection as a whole...
  30. MathematicalPhysicist

    Theories in maths which do not use the axiom of choice.

    i read that there are some logicians who do not use the axiom of choice in their axioms systems. i wonder what is the math that isn't using the axiom of choice, or what theories do not use it?
  31. L

    Is the QM axiom for measurements misleading?

    Was the QM postulate for measurements misleading? Interactions and their understanding is the central topic of QM. So it is for CM too. The interaction between a QM system and a (macroscopic) Classical system is for sure an extremely interresting subject. Particularly when we want to describe...
  32. MathematicalPhysicist

    Axiom of choice and natural numbers.

    i have a few question, that i hope they will answered. 1) let w={0,1...,n,..}={0}UN, and let f:wxw->w such that the next requirements apply: a) f(0,n)=n+1 b) f(m+1,0)=f(m,1) c) f(m+1,n+1)=f(m,f(m+1,n). i need to prove that for every n,m in w, the next statement applies: f(m,n)<f(m,n+1)...
  33. 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...
  34. H

    What is the role of the Axiom of Choice in algebraic structures?

    I was wondering: what is the proof idea behind results such as: (Every vector space has a basis) iff AoC (All bases for a vector space have the same cardinality) iff AoC (Every field has an algebraic closure) iff AoC One direction is obvious, but I have no idea how to begin the other...
  35. M

    Is the Probability Axiom Valid for Mutually Exclusive Events?

    Hi I have this here probability axiom which I'm not sure what I have understood correctly. Let B_1 \ldots B_n be independent events Then P(B_1 \mathrm{U} \ldots \mathrm{U} \ B_n) = 1 which is the same as P(B_1) + P(B_2) + \ldots + P(B_n) = 1 I would like to show that this only is valid...
  36. G

    Undecidable statement made into axiom?

    i am a little bit confused on 1 thing. can a statement that is undecidable in a axiomatic system just be added as a axiom to the original system and never lead to a contridiction? for example godel and cohen showed that the continuum hypothesis is independent of ZFC. does this mean then that...
  37. honestrosewater

    Axiom of Empty Set superfluous in ZF?

    Only some versions of the ZF axioms include an axiom stating that an empty set exists. According to mathworld, the Axiom of the Empty Set (AES) follows from the Axiom of Infinity (AI) and Axiom of Separation (AS), via \exists x (x = x) and \emptyset = \{y : y \not= y\}. I guess they think the AI...
  38. P

    Proving c+1 is an Upper Bound of S with Completeness Axiom

    Let S = \{x | x \in \mathbb{R}, x \ge 0, x^2 < c\} Show that c + 1 is an upper bound for S and therefore, by the Completeness Axiom, S has a least upper bound that we denote by b. Pretty much the only tools I've got are the Field Axioms. I think I'm supposed to do something like: x2 \ge 0...
  39. C

    Dedekind's Axiom: Exploring its Fundamentals & Contradiction

    Not sure if this is the place to ask this. It concerns Dedekind's axiom. Quoting from Dantzig this says: "If all points of a straight line fall into two classes, such that every point of the first class lies to the left of any point of the second class, then there exists one and only one...
  40. C

    Vector Space Basis and the Axiom of Choice: Are They Interchangeable?

    Is the existence of basis in all vector space equivalent to the axiom of choice?
  41. L

    Theorem: Bijections & Axiom of Choice

    If f X---> Y is a function and if there is exactly one function g:Y---> X so that f o g = id_y, the f is a bijection and g=f^-1. Do I need to use the axiom of choice to prove this theorem?
  42. E

    Does R-omega satisfy the first countability axiom?

    Does R-omega satisfy the first countability axiom? (in the box topology)
  43. cepheid

    What is the Completeness Axiom?

    If a nonempty set, S, of real numbers has an upper bound M ( x \leq M for all x in S), then S has a least upper bound b. (This means that b is an upper bound for S, but if M is any other upper bound, then b \leq M .) The Completeness Axiom is an expression of the fact that there is no gap or...
  44. L

    From the Schrödinger equation to the wavepacket reduction axiom

    In the very first pages of "Quantum Mechanics" by Landau & Lifchitz, the measurement process is described as an interaction between a quantum system and a "classical" system. I like this interpretation since any further evolution of the quantum system is anyway entangled with the "classical"...
  45. E

    Axiom of Choice and something I find to not be logical

    I heard something along the lines of when you accept the axiom of choice as true, you can then prove using some abstract set theory that by dividing a sphere, you can divide it and then put it together so that it is bigger than it originally was? Is the math behind this proof difficult? And...
  46. M

    What is the problem with a model of parallel evolution?

    In my less than deep undestanding of the biological sciences I understand there is an Axiom that say that the mechainsim of DNA goes in one direction. The DNA will evolve and change and impose itself on the organism, but the process does not work in reverse. Of course thee was the retrovirus...
  47. J

    Proving Axioms: An Explanation

    I know that a theorem can be deduced from the AXIOMS of a formal system, but I do not know how to prove an axiom. Would you please teach me ? How to prove an axiom ? How did the axiomatic rules become axiom ? I think some statements or formulas are axiomatic, although they seem...
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