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

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1answer
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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 ...
9
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1answer
237 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 ...
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2answers
48 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: ...
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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 ...
4
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1answer
135 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
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1answer
67 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
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2answers
60 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 ...
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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 ...
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0answers
25 views

Can the axiom of induction be replaced by simpler axioms from which it could be deduced and proved? [duplicate]

Can the axiom of induction be replaced by simpler axioms from which it could be deduced and proved? If that's possible does it require significant changes to the other peano axioms?
3
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1answer
43 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 ...
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5answers
559 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
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1answer
113 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 ...
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3answers
141 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
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4answers
141 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: ...
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1answer
59 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 ...
2
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1answer
67 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
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2answers
380 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 ...
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2answers
125 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 ...
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2answers
78 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
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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 ...
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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 $$X=\{x_0, x_1,\ldots\},\text{ with } x_1\in x_0, x_2\in x_1,\ldots$$ ...
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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" ...
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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 ...
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3answers
90 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 ...
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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)} ...
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4answers
972 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: ...
2
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1answer
96 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 ...
3
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0answers
52 views

Does “$(\exists f:A\twoheadrightarrow B)\implies(\exists f:B\hookrightarrow A)$” implies the axiom of choice? [duplicate]

Let $P$ denotes the property that if there exists a surjection from set $A$ to set $B$, then there exists an injection from $B$ to $A$. It's apparent that $P$ can be proved in ZFC. My question is that ...
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1answer
36 views

The existence of the sequence corresponding to some asymptotic sequence

The following proof of the axiom of choice by induction is obviously false: Let $(\Lambda)_{i=1, 2, \ldots}$ be an infinite sequence of nonempty sets. When $i=1$, self-evident. We will assume this ...
3
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1answer
113 views

Models of the successor function

I would like to ask a few questions about models of the succesor function (s(x)=x+1), intact that is a bit vague, consider $T_{S}$ to be the set of axioms given by; S1: $\forall xy[s(x)=s(y) ...
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3answers
134 views

The necessity of the axiom of induction

$\underline{First\ question}$ Let $P(n)$ be a proposition about $n$. In standard mathematical induction, we require: (1)$P(0)$ holds. (2)If $P(n)$ holds, $P(n+1)$holds. Here we use "the axiom of ...
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1answer
63 views

Axiomatizability in monadic second-order logic

For my thesis in finite model theory I'm considering some basic classes of structures, and I want to show in which logical systems they can or cannot be axiomatized. I now consider the class ...
2
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1answer
55 views

Some weaker axiom than “no nontrivial zero divisors.”

I would like to know if there a standard term for or well-known applications of the following axiom for rings or semigroups with zero (which is weaker than the "no nontrivial zero divisors" axiom): ...
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3answers
307 views

Which axiom of set theory does this formula represent ? Why? [closed]

Which axiom of set theory does the statement below represent? Why? \begin{align}\exists x\bigg(&\forall y\Big(\neg\exists z\left(z\in y\right)\to y\in x\Big)\\&\land\forall w\Big(w\in ...
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1answer
26 views

Number of Distinct Axiomatic Systems

This may well be a vaguely formulated question. Please bear with me and help me modify it to make it meaningful and rigorous, or show that it is hopelessly meaningless. I understand an axiomatic ...
2
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0answers
51 views

Lebesgue Measurable Sets & Axiom of Determinacy

While reading some logic theory I bumped against the theorem which states that every set of reals is Lebesgue measurable, assuming the axiom of determinacy. To prove this theorem it apparently ...
5
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1answer
110 views

How can the axioms (and primitives) of Tarski's axiomatization of $\Bbb R$ be independent?

While reading through this Wikipedia page about Tarski's axiomatization of the reals, a particular bit of text jumped out at me: Tarski proved these 8 axioms and 4 primitive notions independent. ...
4
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0answers
107 views

Axioms of Newtonian Mechanics

Axiomatically speaking, could Newton's laws be derived (as theorems) from the conservation of momentum and energy -- along with a few suitable definitions of things like an inertia frame and force? ...
6
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2answers
144 views

If $a=b$ then $a+c=b+c$? [duplicate]

A friend of mine just asked me how to prove that if $a=b$ then $a+c=b+c$, where $a,b$ and $c$ are real numbers, I'm not sure what I should answer. I have a book called introduction to logic and to the ...
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2answers
153 views

Why are Euclid axioms of geometry considered 'not sound'?

The five postulates (axioms) are: "To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line." "To describe a circle with ...
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3answers
75 views

Can a sequence whose final term is an axiom, be considered a formal proof?

Wikipedia gives the following definition of a formal proof: A formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language) each of ...
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0answers
37 views

Order Axioms: '<' or '$\leq$'?

In the axioms given for the real numbers, I see that the order axioms are sometimes given for the '<' relation and sometimes for '$\leq$'. Which is more commonly used these days?
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0answers
86 views

Dependence of axioms of Zermelo set theory

I'm reading a book about Zermelo–Fraenkel set theory, and I was told that there are a lot of dependence relation among the commonly used axiom system (in the question What axioms does ZF have, ...
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1answer
37 views

Proof that if $a > 0$, then $\frac{1}{a} > 0$ using just field and order axioms

This is my attempt to prove that, if $a > 0$, then $\frac{1}{a} > 0$, using just field and order axioms. $$a > 0 \implies a \cdot \left(\frac{1}{a}\right)^2 > 0 \cdot ...
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1answer
35 views

What separates an axiom from a proposition?

I have read that an axiom is defined as "an obvious truth." I have also heard that an axiom is a truth so obvious that no proof could make it more clear. My question is: why is one thing considered an ...
4
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1answer
93 views

Transfinite Induction in Peano Arithmetic

I have heard that Peano Arithmetic (PA) cannot perform transfinite induction up to $\varepsilon_0$. This seems to imply that it can induct up to smaller ordinals, like $\omega$ or $\omega^\omega$ or ...
0
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1answer
42 views

Axiom of Induction Question

I have to use the axiom of induction to prove the summation of k^3 from 0 to n is $(n(n+1)/2)^2$. Here's what I have so far: Let P(n) be the assertion that $0^3+1^3+⋯+n^3=(n(n+1)/2)^2$ Base Case ...
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2answers
66 views

Set of axioms for finite subset of Natural Numbers

I would like to get a set of Peano like first-order axioms for a finite subset of natural numbers $N'$ such that $0 \leq N' \leq Max$, with $Max$ denoting the upper-bound. (So my signature might be ...
2
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2answers
151 views

Why is the Generalization Axiom considered a Pure Axiom?

If $\varphi$ is a formula in a first order language $\mathcal{L}$ and $x$ is a variable that is not free in $\varphi$, then the following is a pure axiom $$\varphi \to \forall x\varphi$$ The ...
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1answer
77 views

Prove $(-x)y=-(xy)$ using axioms of real numbers

Working on proof writing, and I need to prove $$(-x)y=-(xy)$$ using the axioms of the real numbers. I know that this is equivalent to saying that the additive inverse of $xy$ is $(-x)y$ but I am ...