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. 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

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  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

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  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

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  7. M

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  8. M

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  9. T

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  10. J

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  11. T

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  12. L

    Mathematical Axioms of General Relativity

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  13. H

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  14. T

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

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  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

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  17. T

    Question regarding vector spaces and axioms

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  18. micromass

    Anybody else sick of the current math axioms?

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  19. A

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  20. dextercioby

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  21. A

    What are the axioms of algebra?

    Can someone please help me?
  22. M

    Proof using axioms for a field

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  23. rpt

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  24. R

    Proving 2+2=4 Using Field Axioms

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  25. G

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  26. R

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  27. silvermane

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  28. silvermane

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  29. D

    Explore Dependence of Axioms for Rings & Commutative Rings

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  30. D

    How can you prove this using only the ring axioms?

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  31. J

    Proving a+b=b+a Using Ring Axioms

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  32. S

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

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  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?

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  35. S

    Axioms and truth in maths?

    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

    Are Axioms of Math Subjective?

    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)

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  43. A

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

    Axioms & Theorems: What's the Difference?

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  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

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  47. O

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  48. J

    Determine if the space is a subspace testing both closure axioms

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  49. F

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  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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