Mathematics Stack Exchange
Mathematics Stack Exchange

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.

1787 questions
2
votes
0 answers

Tidy Collection of Axioms Used in Mathematics

The power of mathematics relies on its logical representation and it depends on its strict disjunction between axiom and definition. I'd learned lots of definitions while studying mathematics, but still however, axioms are always given only in the…
snapper
  • 589
2
votes
2 answers

What is the purpose of having obvious axioms in mathematics?

Why not simply use right away? Why is an axiom and what is not? Some of the obvious statements are made into axioms whereas other obvious statements aren't. For example, the axiom of choice, arbitrary union axiom, etc. Isn't it so obvious?
Yashas
  • 299
  • 3
  • 18
2
votes
1 answer

To uniquely axiomatize a theory ... meaning?

What does the following mean, exactly? When axiomatizing a theory, it is interesting to ask whether a set of axioms uniquely axiomatize the theory - $\textbf{that is, is it only this theory which makes this set of axoms true?}$ The first sentence,…
Peamo
  • 21
1
vote
1 answer

How do I prove this implication using axiomatic methods?

I am trying to prove that $a >0$ if and only if $\frac{1}{a} >0$, But I am having a lot of trouble doing so. So I would have to prove both sides of the implication. That is $a>0 \implies \frac{1}{a} >0$ and $\frac{1}{a} >0 \implies a>0$. I think…
Taylor Costas
  • 11
1
vote
1 answer

Transitive property of equality and the fundamental nature of algrebra

The fundamental nature of algebra rests on the basic rule that whenever two numbers, variables, or expressions are equal, either one can be replaced at any time by the other one. For example, if we assume that a = b implies that b = a (the symmetric…
Ypnypn
  • 440
1
vote
5 answers

Prove that $\sqrt{2}>1$

I need to prove that $\sqrt{2}>1$, but the initial assumption I am given is that $\sqrt{2}>0$. I have $\sqrt{2}>0$ so $2>0$ (multiply by $\sqrt{2}$ on each side). I don't know what my next step should be. We are supposed to use axioms and what not.…
Jack Armstrong
  • 1,965
1
vote
1 answer

Must axioms be proven to be compatible with other axioms?

As we know, an axiom cannot be proven. It is an assumption that certain properties exist, and then we can learn more interesting properties (theorems) from anything that obeys these axioms. So although you cannot prove an axiom, is there a way of…
David Callanan
  • 395
1
vote
1 answer

How to know when a system of axioms is 'complete'?

Here, I (basically) stated the group axioms as follows. $(xy)z=x(yz)$ $xe=x, ex=x$ $xx^{-1}=e$ In that post, answerers Martin and Ittay were critical of the above list for not including $x^{-1}x=e$, even though it follows from the above three.…
goblin GONE
  • 67,744
1
vote
1 answer

If we want to use proof by contradiction, shouldn't we first proof that a statement is either true or false?

As an example how this could go wrong "This statement is false" We could proof this statement by contradiction as follows Assume the statement is false $\implies$ The statement is false $\implies$ the statement is true. Which is a contradiction,…
Poseidaan
  • 551
1
vote
1 answer

Question on Axiomatic Systems

I'm trying to learn the concept of axiomatic systems. I found an example online which is as follows: Let's consider the following example of an axiomatic system. Axiom 1. Every ant has at least two paths. Axiom 2. Every path has at least two…
ultrajohn
  • 185
  • 6
1
vote
0 answers

Fundamental axiom or theorem for multiplication of an equation by any real number?

Is there a fundamental axiom or theorem stating that if two quantities are equal, multiplying both quantities by the same scalar real number results in two equal quantities? I'm imagining something similar to Euclid's "Common Notions," particularly…
Craveable Banana
  • 156
1
vote
1 answer

Differences in the Field Axioms

I've been trying to study the field axioms in order to eventually go through Spivak's Calculus (I'm not a math major, but just interested). I noticed that different books have different axioms. For example, Spivak lists the order properties as:…
Honus27
  • 21
0
votes
3 answers

Difference between postulates, axioms, and theorems?

I'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the operators they use, the major concepts they deal with…
user1299028
  • 211
  • 2
  • 9
0
votes
1 answer

A very simple question

$vx = z$ $zb = y$ Which means $vxb = y$ Does this build on an axiom (and which)? I have to prove a statement using only some specific axioms. But I don't know if I'm allowed to do that.
Brugerfugl
  • 215
0
votes
1 answer

Is consistency sufficient for existence?

In his Mathematical Analysis I, Zorich says the following after introducing the reals axiomatically: In relation to any abstract system of axioms, at least two questions arise immedi- ately. First, are these axioms consistent? That is, does there…
EE18
  • 1,211
1
2
3 4