# 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.

View More On Wikipedia.org
1. ### A Axiom of choice

This set-theory theorem is very easy to prove: (*) if A≈B & C≈D & A∩C=∅ & B∩D=∅ then A∪C≈B∪D It seems intuitive that if one replaces the strong A∩C=∅ & B∩D=∅ condition by the weaker A∩C≈B∩D the implication (**) if A≈B & C≈D & A∩C≈B∪∩D then A∪C≈BD still holds. (**) does not seem to be much...
2. ### B Is everything in math either an axiom or a theorem?

Considering math as a collection of true/logically consistent statements, I see only two possibilities: either the statement is true and can be proven, which means it's a theorem. Or it's true but cannot be proven, which means it's an axiom. Is there a third possibility? Or maybe more? I feel...
3. ### Prove the 2nd axiom of mathematical logic using the Deduction Theorem

prove: The 2nd axiom of mathematical logic 2) $((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))$ By using only the deduction theorem
4. ### MHB ZFC and the Axiom of Power Sets ....

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focused on Searcoid's treatment of ZFC in Chapter 1: Sets ... I need help in order to fully understand the Axiom of Power Sets and Definition 1.1.1 ... The...
5. ### How Can I Use Axioms to Prove x^2 ≥ 0?

(x)(x)>0 (D) (x+(-x))(x+(-x)) >0 (A4) x^2 + 2(-x)(x) + (-x)^2 >0 (D) x^2 - 2x^2 + (-x)^2 >0 -x^2 + (-x)^2 >0
6. ### I Axiom of Choice: Choosing Identical Objects

From Wikipedia entry on the Axiom of Choice: [1] What about a finite set of indistinguishable things (e.g. identical socks)? Do we need to invoke the axiom? [2] Is there any physical consequence of this axiom, i.e. is there any physics experiment where the calculations to predict the result...
7. ### 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...
8. ### MHB Proving matrix group under addition for associative axiom

Dear Everyone, I have some feeling some uncertainty proving one of the axioms for a group. Here is the proof to show this is a group: Let the set T be defined as a set of 2x2 square matrices with coefficients of integral values and all the entries are the same. We want to show that T is an...
9. ### I Why is the Axiom of Power Set needed?

In the Zermelo-Fraenkel axioms of axiomatic set theory we find: Axiom. Given any set x, there is a set such that, given any set z, this set z is a member of if and only if every element of z is also an element of x. Why is this needed as an axiom? why isn't it merely a definition? Under...
10. ### I Axiom of Choice: Disjoint Family ##\Rightarrow ## Power Set

So apparently the proof involves a trick that converts the problem of a general power set ##\mathscr{P}(M)## of some set ##M## which has of course the property of not having pairwise disjoint set-elements to a problem that involves disjoint set-elements. I do not understand why this trick is...
11. ### A Is irreducibility justified as a Wightman axiom?

My discussion (copying this from a Facebook post dated April 29th, 2018): This weakening of the Wightman axioms is not considered in, for example, Section 3.4 of R F Streater, "Outline of axiomatic relativistic quantum field theory", Rep. Prog. Phys. 38 771-846 (1975), where Streater critiques...
12. ### I Understanding the Induction Axiom: Notation & Equivalence

So , what I was wondering about was a slight difference in notation, for which I am not certain if correct (mine, in particular.). The induction axiom says: If M is a subset of ℕ, and if holds that: a)1∈M b)(∨n∈ℕ)(n∈M→s(n)∈M) then M=ℕ. Now my question is: why do we write (∨n∈ℕ)(n∈M→s(n)∈M)...
13. ### A Let's remove one axiom from Euclidean geometry

I'm wondering what could happen if we remove one axiom from Euclidean geometry. What are the conseqences? For example - how would space without postulate "To describe a cicle with any centre and distance" look like?
14. ### A Need (or not) for invoking axiom of choice in a certain case

I was wondering about the following scenario, we have a certain differentiable manifold with the standard topology not induced by any previous metric structure on the manifold. There is no natural way to identify a vector with its dual(no canonical isomorphism between them), If we had to...
15. ### A Axiom of Choice not self evident?

It seems that it is self evident. After all, a set A consists of things. These things exist as part of the set. Therefore the idea is that I can think about the item of the set without really thinking about the other items in that set. The set is in a sense the totality of the parts. Why is this...
16. ### B Does Every Set in the Axiom of Choice Include an Empty Set?

Following theorems are congruent(a) Axiom of Choice (b) if ∀i:i∈I: <Yi | i∈I > → Yi≠Ø (c) Ø∉S → ∃f: f is on a set S s.t. f(X)∈X for all X∈S. where f is choice function of S. I am confused with the theorem (c), as how the Collection S does not include empty set. I believe every set needs to...
17. ### I Ascending subset sequence with axiom of choice

Is it possible to use Axiom of Choice to prove that there would exist a sequence (A_n)_{n=1,2,3,\ldots} with the properties: A_n\subset\mathbb{R} for all n=1,2,3,\ldots, A_1\subset A_2\subset A_3\subset\cdots and \lim_{k\to\infty} \lambda^*(A_k) < \lambda^*\Big(\bigcup_{k=1}^{\infty}...
18. ### MHB The Completeness Axiom .... and Sohrab Exercise 2.1.29 ....

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.29 ... Exercise 2.1.29 (including the Completeness Axiom) reads as...
19. ### Proving Independence of Fano's Geometry Axiom 4

Homework Statement In Fano's Geometry, we have the following axioms a. There exists at least one line b. Every line has exactly three points on it c. Not all points are on the same line d. For two distinct points, there exists exactly one line on both of them e. Each two lines have at least one...
20. ### B Is Infinity Necessary? Exploring the Role of ZF Axioms in Set Theory

Obviously information exists. From the concept of information quanta we can create physical models of mathematical concepts. Are ZF axioms redundant?
21. ### A Formal axiom systems and the finite/infinite sets

The hereditarily finite sets(a subclass of the Von Neumann universe) are an axiomatic model that corresponds to the usual axioms of set theory but with the axiom of infinity replaced by its negation(showing its independency from the other axioms of set theory). Some mathematicians (a minority)...
22. ### MHB Understanding the Completeness Axiom: A Discussion on Ross' Analysis

I was reading through the early chapters of Ross' book on analysis in the section covering the completeness axiom. See below. Followed by a few examples. I'm confused as to why in the example (e), the set does not have a minimum. I can understand that it does not have a maximum, but it...
23. ### MHB Axiom of Infinity and Garling, Theorem 1.7.4 - the successor set Z^+

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with...
24. ### I Axiom of Infinity & Garling, Th. 1.7.4 & the successor set

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with the...
25. ### 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...
26. ### I Set Theory: Separation Axiom and Garling's 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...
27. ### A proof in the Hilbert-style axiom system

Homework Statement Provide a complete formal proof that ## \vdash ((A \rightarrow B) \rightarrow C) \rightarrow (B \rightarrow C)##. Homework Equations I am only allowed to use modus ponens and these four 'sentential logic' axioms: A1 ## \neg \alpha \rightarrow (\alpha \rightarrow \beta)## A2...
28. ### A Question about axiom of regularity

It has been stated that in axiom of regularity , a set cannot be an element of itself and there is a proof for which S={S} . I can understand his proof since S is the only element and hence its method of proof is viable here . But , what if I change the question to S= {S,b} ( it is a set which...
29. M

### Every positive real number has a unique positive n'th root

Homework Statement Show, using the axiom of completeness of ##\mathbb{R}##, that every positive real number has a unique n'th root that is a positive real number. Or in symbols: ##n \in \mathbb{N_0}, a \in \mathbb{R^{+}} \Rightarrow \exists! x \in \mathbb{R^{+}}: x^n = a## Homework...
30. ### 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...
31. ### I Axiom 10 of Vector Spaces

So I understand how to prove most of the axioms of a vector space except for axiom 10, I just do not understand how any set could fail the Scalar Identity axiom; Could anybody clarify how exactly a set could fail this as from what I know that anything times one results in itself 1u = u...
32. ### A I can't understand axiom of regularity....

It states that no set can be an element of itself..what does it exactly mean?why can't. We consider the whole set as a single element?
33. ### Axiom of QM is Observation... but

It is said the axiom of QM is observation.. but if observation is secondary effect of more primary dynamics that don't involve observations.. could it still be called QM, and what's it supposed to be called? In other words. Say QM is emergent from a deeper reality.. and we are to study the...
34. ### Have I just invented a new axiom?

I doubt it but I was doing some work on trying to remove time from Classical Physics (just for the hell of it) and I came across a formula that made me go "huh, not seen that before, but it's kind of neat." Just out of curiosity has anyone seen this formula before? X = √ ((X/2Π) * (X*2Π))
35. ### Axiom of Pair and Axiom of Union?

So I've been learning Set Theory by myself through Jech and Hrabeck textbook, and I'm having trouble understanding some axioms. 1. Homework Statement What exactly is the difference between the axiom of pair and axiom of union? From what I understood, the axiom of pair tells us that there is a...
36. ### MHB ZFC and the Axiom of Foundation

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Foundation which reads as shown...
37. ### The Rational Plane and Dedekind's Axiom

Homework Statement Here the formulation of Dedekind's axioms that I am using:Suppose that line ℓ is partitioned by the two nonempty sets ##M_0## and ##M_1## (i.e., ##\ell = M_0 \cup M_1##) such that every point between two points of ##M_i## is is also in ##M_i##, for ##i = 0,1##. Then there...
38. ### MHB Understanding ZFC and the Axiom of Infinity: Simple Explanation and Examples

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Infinity which reads as shown...
39. ### MHB Set Theory and ZFC - The Axiom of Replacement - Searcoid, Pages 6-7

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Replacement which reads as shown...
40. ### Exploring the Controversy Around the Axiom of Choice

Hi guys. So I've been wondering, what's so controversial about the axiom of choice? I heard it allows the Banach-Tarski Paradox to work. A little insight would be much appreciated, thanks.
41. ### Is speed of light in vacuum an axiom in relativity?

The speed of light is constant in all frame of reference... So the relative motion of the source of light and the frame of reference from which you are making measurement does not matter... Is it an Axiom in relativity ? I understand that they are experimentally proved concepts... But is it an...
42. ### Comprehension Schema: The Possibility of Multiple Sets with the Same Property

P(x) = x ∉ x ⊃ for any set A, there is a set B such that x ∈ B iff x ∈ A and x ∉ x Does the above mean that different things can bear the same property. For instance, x can be bipedal means x can be an element of the set human or x can be an element of the set ostrich.
43. ### Confusion over Axiom of Choice.

Hello, What "choice" does the Axiom of Choice permit us to make? I've searched high and low and not found a satisfactory answer. To me it seems to add no new information to its hypotheses: Given an arbitrary collection of non-empty sets, isn't it true, without assuming anything and just by...
44. ### Morse-Kelley Class Comprehension axiom and Russell's paradox

As I understand the ZFC solution to Russell's paradox, since {x|x\notinx} must be {x|x\notinx}\capS for some set S, the paradox goes away, but in Morse-Kelley, if I understand Class Comprehension correctly, although again there must be some M such that {x|x\notinx}\capM, this M may be a proper...
45. ### 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...
46. ### Question about the Axiom of Dependent Choice

The Axiom of Dependent Choice, a weaker version of the Axiom of Choice, states that for any nonempty set X and any entire binary relation R on X, there is a sequence (x_n) in X such that x_n R x_n+1 for each n in N. My question is, what would happen if you restricted the relations to...
47. ### Is absorption an axiom for a boolean algebra?

According to wikipedia, absorption is an axiom for a boolean algebra. This seems incorrect to me, since I believe absorption can be proved from the other axioms (distributivity, associativity, commutativity, complement, identity). Thoughts? ## AB' + A = AB' + A*1 = A(B'+1) = A(1) = A ## BiP
48. ### Question about Axiom of Specification

In page 6 of Naive Set Theory by Halmos, he introduces the definition of the axiom of specification, then sets up one example based on the axiom, in which he changes ##S(x)## to ##x \not \in x## to illustrate something. I understand that this mean ##x## doesn't belong in ##x##. Afterwards...
49. ### A countable subset of an infinite set and the axiom of choice.

According to a result of Paul Cohen in a mathematical model without the axiom of choice there exists an infinite set of real numbers without a countable subset. The proof that every infinite set has a countable subset (http://www.proofwiki.org/wiki/Infinite_Set_has_Countably_Infinite_Subset) is...
50. ### The axiom of choice one a finite family of sets.

The axiom of choice on a finite family of sets. I just been doing some casual reading on the Axiom of CHoice and my understanding of the is that it assert the existence of a choice function when one is not constructable. So if we have a finite family of nonempty sets is it fair to say we can...