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

learn more… | top users | synonyms

1
vote
2answers
81 views

Proper definition of addition and multiplication

I recently started to study maths at university and in the analysis course we started, as usual, by looking at the axioms of $\mathbb R$ as a field. I think, I've understood the underlying intuition ...
2
votes
1answer
52 views

ZF: Difference between POW and SEP?

In ZF, the Power Set Axiom (POW) says that given any set $A$, there exists a set $\mathcal{P}(A)$ such that $$ \forall a(a\in\mathcal{P}(A)\leftrightarrow a\subseteq A)\tag{1} $$ Questions: ...
-1
votes
2answers
155 views

Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness?

According to 1 2, the third Kolmogorov axiom is for disjoint sets $(A_n)_{n \in \mathbb{N}}$ $P(\cup_n A_n) = \sum_n P(A_n)$ Is that really disjoint rather than pairwise disjoint? If we ...
7
votes
1answer
146 views

Axiom of determinancy in action

I was wondering if anybody could give examples of actual works which utilized the axiom of determinancy. My issue is that I have heard of it, and read the Wikipedia, but have had trouble actually ...
0
votes
1answer
76 views

Question about a list of ZF axioms and, in particular, SEP

In A.G. Hamilton's Logic for mathematicians, eight axioms of ZF are given: EXT, NULL, PAIR, UNION, POW, REP, INF and REG. The Axiom Scheme of Replacement is formulated like this: $$ (\forall x_1)(\...
0
votes
2answers
70 views

Understanding boolean algebra and boolean axioms?

I'm currently studying discrete mathematics and am having some difficulties with understanding boolean algebra. To be specific, I'm stuck on the following question: ...
-7
votes
1answer
141 views

Gödel's theorem, un-decidable propositions and axioms of a formal theory [closed]

There was a similar question of mine some time ago (which is now closed, but should not be). Anyway the question is similar, but simplified to avoid (mostly) "pedantic" objections, which bypass the ...
0
votes
2answers
46 views

Proof of the associative property of segments (axiomatic geometry)

Show that the sum of the segments is associative, i.e (AB + CD) + EF = AB + (CD + EF). P.S.: Here we don't speak in congruente segments, but in equal segments. P.S.²: I don't even know how to start ...
11
votes
4answers
547 views

Kolmogorov's probability axioms

Why Kolmogorov's axioms are considered such a breakthrough in probability theory? They are just 3 simple statements everyone can agree with. When creating a system of axioms like this it's necessary ...
1
vote
3answers
96 views

Defining the constructible universe without model theory

The constructible powerset is defined in Wikipedia as: $$\operatorname{Def}(X) := \Bigl\{ \{ y \in X \mid (X,\in) \models \Phi(y,z_1,\ldots,z_n) \} \Big| \Phi \text{ is a wff and } z_{1},\ldots,z_{n}...
35
votes
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 n:0\...
-1
votes
2answers
123 views

When is equality not reflexive? [closed]

In what fields, rings, groups, etc, is equality not symmetric? i.e. $$x=y$$ but $$y\neq x$$ I have never heard of such a field/ring/group that exhibits this, but I know little about such things. ...
0
votes
0answers
78 views

Why are definitions written as 'if-then' statements instead of 'if-and-only-if' [duplicate]

An example from Rudin would be: (c) if $x + y = 0$ then $y = -x$. There may be times when one would have to use the fact that since $y = -x, x + y = 0$. While this is fairly intuitive, professors ...
26
votes
3answers
628 views

Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: Commutativity ...
0
votes
1answer
52 views

Couple of questions on the Axiom of Extensionality

I understand that the axiom of Extensionality says that two sets are equal iff they have the same elements. Which is clear enough. But take a look the definition for this axiom given by mathworld(http:...
1
vote
3answers
66 views

Axioms for Group Action

Reading the Wikipedia article on group action, I am wondering, why are the axioms stipulating that a group action obey both "compatibility" and "identity"? If a group action is merely a group ...
0
votes
1answer
34 views

Continuum hypothesis and non measurable set

This is from Chap 8 of Real and Complex analysis of Rudin. The author does not present a proof (using the continuum hypothesis) for the existence of the function $j$. Where can I find such a ...
0
votes
2answers
32 views

CC for finite sets and equivalent condition

If we assume for all sequence of sets od cardinality exactly 2, there exist choice function. Can we prove Countable choice for finite sets
0
votes
0answers
18 views

CC and one of its equivalent condition [duplicate]

How we can prove: CC is equivalent to for all sequence(Xn) of non empty sets there exist (xn) which meets infinitely many (Xn) One side assuming CC is obvious. But what about converse. How we can ...
1
vote
1answer
123 views

Proof of identity A = A or 1 = 1

Is 1 = 1 an assumption? I feel it's a very good assumption, but is there a proof for it? Imagine a world where people were contesting it, where equivalence wasn't a common sense concept. In reality no ...
2
votes
1answer
35 views

Use the axioms for the field of real number to prove $1/(-a) = -(1/a)$

How would you prove using the axioms for the real numbers that $1/(-a) = -(1/a)$? I tried the following: $1/(-a) = 1/(-a) + 0(1/a) = 1/(-a) + (1+(-1))(1/a) = 1/(-a)+1/(a) + (-(1/a))$ But I don't ...
4
votes
2answers
428 views

Is this axiom self-contradicting?

I was on physics stackexchange and came across an unusual answer where it was stated that the axiom, $$\forall x ((x \in x) \land (x \notin x))$$ Creates an axiom system where "nothing" exists in it....
2
votes
0answers
29 views

Multiple order axioms independence [duplicate]

Let $T$ be a theory, let $L$ be its language, let $A$ be its set of axioms and let $P_0 \in L$ be a property. $P_0$ could be : Consequence of $A$ The negation of a consequence of $A$ Independent of ...
1
vote
1answer
63 views

What is a topology minus the axiom that $\varnothing \in \tau$.

For instance, this is the case with defining $U \subset \Bbb{N}$ to be open iff $\sum_{x \notin U} \frac{1}{x} \lt \infty$ if we let $\sum_{x \notin \Bbb{N}} = 0$. I can't seem to get $\varnothing$ ...
3
votes
1answer
268 views

Set Theory ZF Axioms Doubt

I have a pretty basic question about the symbolic representation of the axiom of extensionality for set theory, which states that $$ \forall A \forall B [ \forall x (x\in A \iff x\in B)] \iff A = B $...
10
votes
1answer
247 views

The theory in probability

Consider a real-life experiment (perhaps written as a problem in a textbook): A coin is continually tossed until two consecutive heads are observed. Assume that the results of the tosses are mutually ...
0
votes
2answers
49 views

Is this set of event axioms complete?

In the notes' first chapter of the course "Discrete Stochastic Processes" presented by Prof. Robert Gallager, the "Axioms for events" defined as follows: 1.2.1 Axioms for events [Chapter 1: ...
0
votes
0answers
83 views

A simple question about rational numbers without a simple proof?

As in this question, study the quasigroup $(Q_+,/)$ of positive rational numbers under division. There are two obvious identities: $a/(b/c)=c/(b/a)$, for all $a,b,c\in Q_+$ $(a/b)/c=(a/c)/b$, for ...
6
votes
1answer
204 views

Can the generalized continuum hypothesis be disguised as a principle of logic?

A cool way to formulate the axiom of choice (AC) is: AC. For all sets $X$ and $Y$ and all predicates $P : X \times Y \rightarrow \rm\{True,False\}$, we have: $$(\forall x:X)(\exists y:Y)P(x,y) \...
7
votes
1answer
70 views

A familiar quasigroup - about independent axioms

A quasigroup is a pair $(Q,/)$, where $/$ is a binary operation on $Q$, such that (1) for each $a,b\in Q$ there exists unique solutions to the equations $a/x=b$ and $y/a=b$. Now I want to extract a ...
3
votes
2answers
72 views

Alternate Axiom of Infinity

The Axiom of Infinity states that there is a set $S$ containing $\varnothing$ such that if $x$ is an element of $S$ then so is $x\cup\{x\}$. Is the following variant equivalent? There exists a ...
5
votes
3answers
1k views

How can the axiom of choice be called “axiom” if it is false in Cohen's model?

From what I know, Cohen constructed a model that satisfies $ ZF\neg C $. But if such a model exists, how can AC be an axiom? Wouldn't it be a contradiction to the existence of the model? Only ...
3
votes
1answer
47 views

Algebraic structures and axiomatic systems

In one textbook appears the following sentence: An algebraic structure is a nonempty set $M$ together with one or more operations (i.e. a function $*:M\times M\rightarrow M$) which satisfy some ...
10
votes
5answers
624 views

Foundation of ordering of real numbers

This might be a silly question, but what is the mathematical foundation for the ordering of the real numbers? How do we know that $1<2$ or $300<1000$... Are the real numbers simply defined as ...
3
votes
1answer
117 views

Does the Russell Set exist?

I am currently reading "Naive set Theory" by Paul Halmos. In the second chapter, on the axiom of specification we show that the Universal Set does not exist. The proof is the following: Lets ...
1
vote
3answers
155 views

What is the definition of axiom (mathematically speaking)

I'm wondering about the exactly definition of axiom (mathematically speaking). The definition of this term seems a little blur in my mind. For example, the definition of point in the Euclidean ...
2
votes
4answers
147 views

$A=\{A,\emptyset\}$ and axiom of regularity

The axiom of regularity says: (R) $\forall x[x\not=\emptyset\to\exists y(y\in x\land x\cap y=\emptyset)]$. From (R) it follows that there is no infinite membership chain (imc). Consider this set: $...
1
vote
1answer
60 views

The order isomorphism theorem without Replacement

In $\sf ZF$, there is the following theorem: Theorem: For any well-ordered set $\langle A,\prec\rangle$, there is a (unique) order isomorphism $F$ from $\langle\alpha,\in\rangle$ to $\langle A,\...
2
votes
1answer
68 views

Are there unsolved problems known to be not independent of the axiomatic system it is proposed in?

Are there unsolved problems known to be not independent of the axiomatic system it is proposed in? For example, is Goldbach's conjecture known to be provable using the axioms of PA? I believe I ...
3
votes
2answers
479 views

Difference between Logical Axioms and Rules of Inference

What's the difference between Logical Axioms and Rules of Inference? In my understanding, both are ordered pairs of formulas which are used to reach a conclusion through syllogisms. My questions ...
6
votes
2answers
129 views

How do we know that our definitions/axioms are not contradictory?

Let us assume that we have declared some axioms. Now, we wish to declare a new axiom too. How do we establish that the new axiom is not a consequence of, nor a contradiction to the original set of ...
0
votes
2answers
84 views

Why is the axiom of pairing needed in Von Neumann-Godel-Bernays Set Theory?

Why do we need the axiom of pairing in Von Neumann-Godel-Bernays Set Theory? Doesn't the following prove it? Suppose $A$ and $B$ are sets. Using the axiom of class comprehension, we can form a ...
2
votes
1answer
55 views

“Every set is equinumerous to a well-founded set” - provable in $\sf ZF-Reg$?

What is the relationship of the following to other axioms of $\sf ZFC$? $\sf WB$: Every set $A$ is in bijection with a set well-founded by $\in$. Obviously, $\sf ZF$ implies $\sf WB$ (because ...
0
votes
2answers
54 views

Axiom of foundation allows sets consisting of a descending sequences plus some 'atom'?

I am reading a text which describes how the Axiom of Foundation prevents sets that are built from a descending sequence such $$X=\{x_0, x_1,\ldots\},\text{ with } x_1\in x_0, x_2\in x_1,\ldots$$ ...
3
votes
0answers
83 views

What do the ZFC axioms look like in terms of subset?

I'm reasonably familiar with the ZFC axioms. I know they can be formalized in many different ways. However, I usually see them presented in terms of the elementary "propositional calculus primitives" $...
31
votes
4answers
2k views

Axiomatic definition of sin and cos?

I look for possiblity to define sin/cos through algebraic relations without involving power series, integrals, differential equation and geometric intuition. Is it possible to define sin and cos ...
1
vote
3answers
95 views

Why do we have Axiom of Pairing but we don't have its generalisation, i.e a collection exists instead of pairing axiom?

For unions we have the generalised axiom, not just union for pairs. But for pairing we don't have generalisation, that a collection exists for any number of sets. If we have some axiom that says ...
1
vote
0answers
82 views

is axiom of powers required?

we can form any subset of a set using axiom of specification. then we can form collection of subsets using axiom of pairing repeatedly. we have for every condition S , (A={x belongs to X & S(x)} =...
15
votes
4answers
1k views

Vector Spaces: Redundant Axiom?

Question Why are the axioms for vector space independent? More precisely $1x=x$ seems redundant... (I take the axioms from: Wikipedia) Explanation One has for zero vector: $$\lambda0+\lambda0=\...
2
votes
1answer
103 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 ...