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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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What is an axiom in layman's terms?

I tried to explain to my kids what is an axiom in very nontechnical terminology (also known as layman's term) but I could not find anything that they could relate to easily. Do anyone have a good explanation?
Andy K
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Multiplication by One

Throughout school we are taught that when something is multiplied by 1, it equals itself. But as I am learning about higher level mathematics, I am understanding that not everything is as black and white as that (like how infinity multiplied by zero…
esote
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Has anyone ever proposed additional axioms?

According to Wikipedia, Godel's incompleteness theorem states: No consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all facts about the natural …
Casebash
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What constitutes an axiom - Spivak Calculus ch. 1

In chapter 1 of Spivak's Calculus text he lays out some fundamental axioms of the integers. For instance that: $a \cdot 1 = a$, for all $a$. However he doesn't list an axiom that for instance says: $a \cdot 0 = 0$, for all $a$. This seems a bit…
Joseph P
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Stephen Wolfram on axiomatic systems?

I was watching a video where Wolfram was discussing the development of Mathematics, he said something along the lines of: " There is a whole universe of possible mathematics. I was curious about this question for logic for example. We always think…
user4568
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Set is a structure.. or structure is a set.., which is correct?

A mathematical structure is a set (or sometimes several sets) with various associated mathematical objects such as subsets, sets of subsets, operations and relations, all of which must satisfy various requirements (axioms). The collection of…
Arshad Nazeer
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What exactly is an axiom, and can an axiom be proven?

I've come here after reading about proving SAS congruence, where the response I read was about how it was an axiom. I'm not sure, however, what exactly is an axiom. And can it be proven? And if it can't be proven, how can one just assume it true?
Ethan Chan
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Axiom - what does this statement about the “truth” of axioms mean?

When reading the Axiom page on Wikipedia, I encountered this sentence at the end of the lead: Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is an open question[citation needed] in…
user392768
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Is the parallelogram law a theorem or an axiom?

I'm learning about inner product spaces and I am able to prove it within an inner product space. Is this a theorem or an axiom in euclidean geometry?(note: not the geometry of Descartes)
shooting-squirrel
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What is the rule for something divided by itself equaling 1?

Is there a name for the mathematical rule/axiom/property $x/x = 1?$ What are the conditions for it to apply? For instance, the rule does not apply where $x = 0$ or $x = \inf$. I saw one site that claimed it only applied to real numbers, but it does…
Mike Fairhurst
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Are there doubtful axioms?

Over my life, I've encountered three different definitions for mathematical axioms: Axioms are statements that must be accepted on faith. Unbelievers shall be punished by eternal damnation grade reduction or course failure. Axioms are statements…
Robert Columbia
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Are there examples of theorems proved via proper (i.e. non-conservative) extensions?

This is not a question about set theory specifically, but lets talk about ZFC just for concreteness Suppose we have a sentence $\phi$ in the language of ZFC, and a proof that either $(\mathrm{ZFC} \vdash \phi) \vee (\mathrm{ZFC} \vdash \neg\phi)$.…
goblin GONE
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Is there a consistent axiomatic system where $1+1 = 2$, $2 \not = 1+1$?

Is it possible to create a consistent axiomatic system where $1+1 = 2$, $2 \not = 1+1$?
Physiks lover
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process of coming up with a definition or axiom?

I am not a maths student but am very interested in the mathematical processes. A major part of mathematics is proofs. The steps carried out on the proofs are all based on the initial axioms or definitions. So my question is, are the initial axioms…
kofhearts
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Fundamentals of mathematics

When I was a teenager, I always thought that mathematical results/theorems constitute absolute truths. However after having studied maths in college, I’ve came across axioms, and things like the continuum hypothesis. When I first read that the…
ml-enthusiast
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