# 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. ### 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. ### 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...

6. ### 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. ### 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. ### 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...

13. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

47. ### 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. ### "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. ### 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. ### 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...