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

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1answer
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How do you formally state the axiom of constructibility?

The Axiom of Constructibility states every set is constructible. My question is, how would one formally state this, since it seems to involve quite a bit of metamathematics? In particular, if one ...
2
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1answer
46 views

An infinite set of axioms in ZF? What does that mean?

Before write this question, I lookeded around enough in this forum for a possible answer and although there are many similar questions, I couldn't find one answer which understand or satisfies me. I ...
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3answers
34 views

Why isn't there a need to specify a single $x$ in the axiom schema of replacement? [on hold]

The axiom schema of replacement needs a function, defined by ${\forall}y({\exists}x:({\forall}z(P(y,z){\iff}(x=z))))$, where $f(y)=x$. My question is: why isn't ${\exists}$ replaced by ${\exists}!$? ...
0
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1answer
13 views

How can I show a set B with 8 elements and two operations (huntington axioms)

How can I show a set B with 8 elements and two operations, such that the axioms of huntington for boolean algebra holds? I found it with set of 2 elemtnts. but can't understand how to start with 8 ...
7
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1answer
76 views

Is there a category theory notion of the image of an axiom or predicate under a functor?

Let me first state that I am a category theory novice so your patience is appreciated. I might be making some very basic conceptional mistakes and I might just need a simpler language to do what I ...
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1answer
55 views

Axiom in Foundations, Extensionality

In my Foundations of Mathematics Textbook I encountered the following problem. The book states that for the domain of discourse $D = \{a,b,c\}$ and binary relation defined as $E = \{(a,b),\, (a,c)\}$ ...
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6answers
1k views

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
72 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) ...
2
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2answers
36 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 ...
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1answer
29 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
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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 ...
2
votes
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 "...
2
votes
1answer
500 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
25 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 ...
16
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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 ...
51
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10answers
13k views

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 ...
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0answers
13 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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1answer
20 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
72 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 ...
2
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3answers
67 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
32 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 ...
1
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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?
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 ...
0
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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 ...
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2answers
94 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, ...
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1answer
96 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 ...
0
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1answer
29 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
46 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 ...
0
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1answer
50 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 ...
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0answers
108 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 ...
38
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3answers
4k views

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 ...
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3answers
92 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
54 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
1k views

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 ...
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0answers
65 views

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 ...
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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1answer
54 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
95 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 ...
0
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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 ...
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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 ...
4
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1answer
158 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. ...
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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 ...
0
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0answers
39 views

How do I prove the pairing axiom by the other ZFC axioms? [duplicate]

To prove the pairing axiom I thought about using the replacement axiom but I do not really know how. Can someone show me how it is done?
4
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2answers
74 views

Axioms of Trigonometry

On Wikipedia it gives a picture of all trigonometric functions of an angle laid atop the unit circle, 1. Obviously there are other trigonometric identities, but what I'm wondering is, does ...
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3answers
36 views

What's a model in axiomatic theory?

In the first answer here: Kolmogorov's probability axioms It is wondered whether there is only one model of the axioms (up to an isomorphism). Could somebody explain this concept? What's a ...
4
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1answer
102 views

A possible alternative to the Axioms of Pair, Union, Infinity and Replacement

In this question we assume that all formulae are in the language of $\sf ZFC$ and that $\sf ZFC$ is consistent. Recall that we say that a formula $\varphi(x,y)$ represents a set-like class relation ...
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1answer
25 views

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 ...