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

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Meaning of the word “axiom”

One usually describes an axiom to be a proposition regarded as self-evidently true without proof. Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises ...
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2answers
67 views

Deducing $((\neg a \to \neg b) \to ((\neg a \to b) \to b)))$ from axioms

I have seen many questions here, using a different set of axioms than mine. Here is mine : $$1) (a \to (b \to a))$$ $$2) ((a \to (b \to c)) \to ((a \to b) \to (a \to c)))$$ $$3) ((\neg b \to \neg a) ...
3
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3answers
233 views

Why is this contradiction using axiom of constructibility incorrect?

Today I thought of this paradox and I'm trying to find the wrong assumption that causes it. Does anyone know what is wrong in the following argument: let $$A=\{x\in \mathbb{R}|\exists\phi:\forall ...
2
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2answers
30 views

Propositional calculus axiom the other way around

I have the following axioms of propositional calculus (as well as modus ponens and the deduction theorem if needed): $$(a \to (b \to a)) \tag1$$ $$ (((a \to (b \to c)) \to ((a \to b) \to (a \to c))) ...
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0answers
53 views

Why is the area of a rectangle given as $l\times h$? [duplicate]

While this may seem like a simple question, I have found it very difficult to answer. My question is "Why is the area of a rectangle given as $l\times h$?" We define area as the "size of a two ...
1
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2answers
105 views

Proof that the minimal inductive set does not contain any more elements than $\{\emptyset, s(\emptyset), s(s(\emptyset)),s(s(s(\emptyset))),…\}$?

I am studying axiomatic (ZF) set theory and in particular the construction of the natural numbers from axiomatic set theory. Let me start by giving some introduction to my question. The axiom of ...
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1answer
25 views

Is the set of all limit ordinals a set? [closed]

It could be a limit ordinal too, then would be a class, but I'm not sure about is that an ordinal number
17
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3answers
1k views

Why does what I've written fail to define truth?

I stumbled across a set of axioms for first order logic a bit ago. Intrigued, I decided to try to write it all down and organise what I read. After I did that, it seemed to me as though one could ...
58
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9answers
5k views

Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
49
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10answers
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Has there ever been an application of dividing by zero?

Regarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and ...
2
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1answer
38 views

What does the word “Comprehension” mean in the Axiom of Comprehension?

I understand roughly what the Axiom of Comprehension means, that any predicate can be used to construct a set of the elements that satisfy the predicate. But in English terms, where does the word ...
16
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2answers
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Proofs given in undergrad degree that need Continuum hypothesis?

Or alternative you need to assume CH is false. I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non ...
2
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1answer
494 views

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 ...
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0answers
24 views

Applications of the Axiom of Regularity to non-set-theoretical Mathematics [duplicate]

In the beginning of my mathematics studies at university, we have learnt that nearly all of ordinary mathematics not dealing with proper classes can be formalized within ZFC, which is a famous ...
14
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11answers
3k views

What is an axiom in layman's terms? [closed]

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 ...
5
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4answers
7k views

What is the difference between an axiom and a postulate?

I hear about axioms in set theory and postulates in geometry, but they seem like the same thing. Do they mean the same thing but then are used in different instances or what? Is one word more ...
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11answers
3k views

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall ...
3
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3answers
325 views

Defining the integers and rationals

What axiomatic system is most commonly used by modern mathematicians to describe the properties of the integers and the rationals? Properties like $a+0=a$, $a*1=a$, $a+b=b+a$, Also given these ...
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0answers
8 views

Scotts axiom, representation theorems for Qualitative -Numerical Probability function relations

Scotts theorem/axiom and other representation theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, ...
0
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0answers
14 views

How to derive the five-segment axiom of Tarski's geometry from Hilbert's axioms?

We are trying to prove that in an arbitrary Hilbert plane (assuming Hilbert's axioms of Group I:Incidence, II:Order and III:Congruence https://en.wikipedia.org/wiki/Hilbert%27s_axioms), Tarski's ...
4
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1answer
69 views

Necessity of being rigorous

Disclamer I am no serious mathematician, just curious Context I recently discovered some set theories, ZFC and IZF in particular. It made me realize that I've studied math a whole year without ...
3
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2answers
595 views

Definition: Mathematical way to define the “left” and the “right”

How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has ...
2
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3answers
62 views

Axioms of Geometry?

I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it ...
2
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3answers
31 views

Inner Product proof, axiom 1

I hate to argue that my text book is wrong. That being said, I am going to try and do just that. The text book says that this IS a valid inner product, I disagree. The vectors u and v are defined ...
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3answers
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Is the Bourbaki treatment of Set Theory outdated?

On page 5 of the following write up, the author asks why the Bourbaki did not notice that their system of Zermelo set theory with AC was inadequate for existing mathematics. Throughout the rest of the ...
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1answer
42 views

Prove axiom of infinity from ZF$-$Infinity +Zermelo's version of infinity

Assume ZF-infinity and assume that there is a set $X$ such that $\varnothing\in X$ and for all $t\in X$, $\{t\}\in X$. How to deduce axiom of infinity from these new axioms?
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2answers
56 views

Choice function without AC in special case [duplicate]

I read the Jech, Set theory, and saw following proposition. (☆) If S is a finite family of nonempty sets, existence of choice function of S can be proved without axiom of choice. I tried to prove ...
2
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1answer
36 views

Requirements for Diagonal Lemma

What are the axioms required for a formal system to be able to state the Diagonal Lemma or Fixed Point Theorem? If possible, could you please also relate with the following systems, if it applies to ...
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2answers
86 views

Proof using only field axioms

Prove if $x, y ∈ R$ and $xy > 0$ then either $x > 0$ and $y > 0$, or, $x < 0$ and $y < 0$ using only the field axioms. These include the Field axioms for addition, multiplication, ...
0
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0answers
106 views

What kind of Properties are Allowed in the Schema of Comprehension?

Studying an introduction to set theory, we were presented with several axioms as the Axiom Schema of Comprehension and the Axiom of Infinity. This last aximom allows the definition of the inductive ...
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1answer
91 views

What book on Set Theory is best to understand motivation for axiomatization?

I am Master of Science in ICT, and I had always been in loved in math. On University we haven't been doing any Foundational Mathematics, the closest being Automata Theory and mention of Church-Turing ...
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3answers
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Can proof by contradiction 'fail'?

I am familiar with the mechanism of proof by contradiction: we want to prove $P$, so we assume $¬P$ and prove that this is false; hence $P$ must be true. I have the following devil's advocate ...
0
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1answer
26 views

Equivalence of Taylor series and its corresponding function and Axiom for infinite summation

Given a function $f(x)$ with a taylor series expansion, is it valid to say that $$f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}f^{(n)}(a)(x-a)^n$$ for all values of x irrespective of whether the taylor series ...
0
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1answer
48 views

Simple proof using only field axioms

I need to prove that $(-x)(-y) = xy$ using only the field axioms. I tried starting with $ since$ $(-(-x)(-y)) + (-x)(-y) = 0$ by A6, or the additive inverse. And then adding $xy$ to both ...
0
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1answer
37 views

Real analysis using field axioms to prove

Prove that (-x)(-y)=xy using field axiom theorems My attempt i know that -(xy) is the additional inverse for xy, and i know xy + -(xy) = 0 (equation 1) i assumed that (-x)(-y)=xy and substitute ...
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3answers
91 views

In relatively simple words: What could be a model of $\sf {ZFC}$?

An $\text{interpretation}$ of a theory consists of A domain of discourse $\mathcal U$, usually required to be non-empty. For every constant symbol, an element of $\mathcal U$ as its ...
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2answers
44 views

Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
0
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1answer
46 views

What axiom in Math says “similar inputs should yield similar outputs”?

It is easy to take for granted the simple idea that similar input $x$ to a function $f(x)$ should yield similar outputs - such that if the difference between $x$ is arbitrary small, then we should ...
14
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8answers
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Zero vector of a vector space

I know that every vector space needs to contain a zero vector. But all the vector spaces I've seen have the zero vector actually being zero (e.g. $\mathbf{0}=\langle0,0,\ldots,0\rangle$). Can't the ...
0
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1answer
29 views

How to verify that $P_n(F)$ is a vector space over field $F$?

I assume $p(x)=a_nx^n+...+a_0$ where $a_i\in F$ and $x$ is variable. To verify it as vector space, I think we need to check axioms. i.e. $\forall x, y,z\in V,\ (x+y)+z=x+(y+z)$ and $\forall x, y\in ...
2
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1answer
97 views

Left and Right hand associativity equivalent first order logic

Say we are in a first order theory, and one of our inference rules is the associative rule saying that we can infer $(A \vee B) \vee C$ from $A \vee (B \vee C)$. Using the other logical inference ...
4
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1answer
148 views

Two congruent segments does have the same length?

The answer to the question in the title seems an obvious ''Yes by definition !''. And this really is the definition from Wikipedia: Two line segments are congruent if they have the same length. ...
4
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4answers
127 views

Why is the infinite set from the axiom of infinity the natural numbers?

Why is the infinite set from the axiom of infinity the natural numbers? Is there any reason such set was chosen? Couldn't the axiom yield a set that looks like $\Bbb R$ for example?
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0answers
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Consequences of the Fact Every Uncountable Set of Reals Has Perfect Set Property

I have recently been working through the consequences assuming the Axiom of Determinacy has and came across the fact that this axiom implies that every uncountable set of reals has the perfect set ...
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3answers
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The existence of the empty set is an axiom of ZFC or not?

I found in the Wolfram MathWorld page of the Axiom of the Empty Set that this is one of the Zermelo-Fraenkel Axioms, however on the page about these ZFC Axioms I read that it is an axiom that can be ...
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1answer
53 views

Axioms of noncommutative rings [duplicate]

Is there an abelian group $R$ with multiplication operator with this properties ? (i) $a(bc)=(ab)c$ (ii) $a(b+c)=ab+ac$ , $(b+c)a=ba+ca$ And a unique element $e$ s.t (iii) $ea=a\quad ...
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0answers
89 views

How can I learn Math intuitively? [closed]

I am currently a Junior in High School. I am in an Intermediate Algebra class, but my teacher does not always explain things in a way I can understand. I like to learn Math intuitively, but my teacher ...
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0answers
37 views

I'm stuck on a proof involving the of Axiom for multiplicative inverses and modular arithmetic.

I am trying to show that the axiom of multiplicative inverses holds on sets of integers modulo P when P is prime. i just need to show that for any non zero integer, n less than P there is a unique ...
0
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0answers
22 views

Can you characterize dimension in terms of the touching axioms?

One alternative axiomatization of topology is the touching axioms. It's basically the same as the closure axioms, and is in fact equivalent to the open set definition. Can we define dimension in a ...
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0answers
38 views

Probability without second axiom (unit measure)

I'm working with functions (namely, representing incoherent degrees of belief) which resemble probabilities, but are actually, say, quasi-probabilites: their values on atomic events (here: atomic ...