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Questions tagged [axioms]

For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.

An axiom is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.

Axioms define and delimit the realm of analysis. In other words, an axiom is a formal statement that is assumed to be true. Therefore, its truth is taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other (theory and domain dependent) truths. An axiom is defined as a mathematical statement that is accepted as being true without a mathematical proof.

It should be mentioned that in modern times some statements receive a status of axioms, but they are still provable from weaker theories using other statements. One famous example is the axiom of choice, which is provable from ZF set theory if we assume Zorn's lemma. Generally, in modern foundations of mathematics, an axiom is just a statement in the base theory.

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Proving that if $a^3=b^3\rightarrow a=b$ using only field axioms.

How can I prove that if $a^3=b^3\rightarrow a=b$ using only field axioms $\forall a,b\in\mathbb{R}$. I feel like it has something to do with $$ a^3-b^3=0\rightarrow a^3+(-a^3)=0 \rightarrow -b^3=(-a^3) $$ but I have no clue how to continue the…
jc426
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Is there an area of mathematics about the study of systems of laws?

To clarify the question, by "systems of laws" I mean something like the collection of physical laws that govern nature, rules of a game like soccer or chess, or a power system that governs what fictional characters can do in a novel/video…
Chidi
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Axioms in Mathematics

Quote from Sir Edmund Taylor Whittaker from his essay in the symposium “What is Science?” written in 1955. He was discussing non-Euclidean geometry and the role of the parallel postulate in Euclidean geometry that led to the discussion of axioms in…
Paul Ash
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Axiomatic system and Proof for axioms

So I am told by a friend that "axioms in an axiomatic system cannot be proved within the axiomatic system". I was wondering how true this is. Is there any actual mathematical theorem that states something like this. EDIT: Along the same lines, how…
TheImperfectCrazy
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Are abstract definitions in math still subject to our understanding of concrete objects? To what extent?

It seems like in mathematics there are two separate meanings of the word axiom: One is like the real or natural numbers, whose axiomatization is based off on concrete objects. The other is like with the definition of a metric space. The is no…
Pineapple Fish
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Is it possible to prove the inconsistency of an axiomatic system?

As I understand it, it is possible to prove the consistency of a given axiomatic system using a stronger axiomatic system, but no system can be proven to be absolutely consistent (essentially, the consistency of the given axiomatic system is…
Art
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What are the axiomatic systems generally used?

When we apply mathematical theorems to perform some computation or to prove some statement, we are relying on an axiomatic system. I am currently finishing computer science studies which include a lot of mathematics but I feel I lack of the origin…
user183748292
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Verifying axiom of substitution?

In Tao's analysis volume 1, I am introduced to this thing called the axiom of substitution. While constructing real numbers from rationals, he defined reals to be formal limits of Cauchy sequences of rationals. He said $\lim a_n=\lim b_n$ iff…
Not Euler
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Minimum eligibility to become a theory

Is there any mathematical theory without any axioms? Theories such as set theory, number theory etc., all has axioms in it. I have confusion between mathematical theory and the word theory in usage such as Automata theory.
hanugm
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axiom of extensionality word definition

I was reading about the axiom of extensionality and in words it reads "If A and B are sets such that for every element x, x is a member of A if and only if x is a member of B, then A is equal to B" am i right in saying that this is not actually…
Carlos Bacca
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Axioms as agreed in the book "Math Proofs Demystified"

I am reading this book "Math Proofs Demystified" by Stan Gibilisco in which the author treats all of the congruence criteria/theorems of triangles as axioms instead of treating of one of them as axioms as using it to prove others as theorems, not…
MathDumbo
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Closure axioms, and it the sets satisfy them

So pretty basic question http://prntscr.com/jcz64c So I am going through the answer, and I check mine, and I got the first one correct, but I cannot figure for the life in me why the second one fails the two closure axioms help really appreciated.…
mathlover
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Axiom of Choice --- Why is it an axiom and not a theorem?

My question is only indirectly about the axiom of choice, I just happened to come to the question via the axiom of choice and will use it to illustrate my problem. So far I thought axioms were propositions that were a) asserted to be true and b)…
user3578468
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Additive and multiplicative identites help

our new maths teacher (BSc Hons) just started a topic. He wrote all these theorems but I'm unable to understand them and prove the statements. They are as follows: Theorem: The addition axioms for field imply the following statements: If x+y = x+z,…
Lord Asbaat
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Contradictory axioms

¿Do you know if there is any mathematical theory (or axiomatic theory) whose axioms, or at least one of them, contradict with other axioms of another theory? Thanks for reading.
Eduardo
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