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

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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
89 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 ...
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4answers
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$A=\{A,\emptyset\}$ 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
38 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 ...
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1answer
49 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 ...
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2answers
71 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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109 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
58 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 ...
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1answer
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“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
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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
71 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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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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62 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
71 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
751 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
49 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 ...
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0answers
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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
25 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 ...
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1answer
79 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
80 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
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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 ...
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1answer
49 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
155 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
22 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 ...
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0answers
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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 ...
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1answer
86 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. ...
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0answers
48 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? ...
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2answers
122 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
66 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
49 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
30 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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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
26 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
27 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 ...
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1answer
70 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 ...
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1answer
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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
65 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 ...
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2answers
76 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
51 views

Axioms Conditional Probability

I am in a game theoretic framework in which I need to allow an agent to be able to conditionalize on events with probability zero. This means that I can not use the classical definition of conditional ...
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1answer
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with Tarki's axioms of geometry what is a plane?

Reading about Tarski's axioms of geometry https://en.wikipedia.org/wiki/Tarski%27s_axioms I was puzzeling how they would look for 3 dimensional geometry. the big problem then is the upper dimension ...
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1answer
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How to formulate the hyperbolic parallel postulate for more than dimensions?

To formulate the hyperbolic parallel postulate for the hyperbolic 2 dimensional (plane) is easy: Given any line ''L'' and point ''P'' not on ''L'', there are at least two distinct lines passing ...
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3answers
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Commutativity of scalar/vector product: $a\mathbf{v}=\mathbf{v}a$ for all $a \in F$ and $\mathbf{v} \in V$

There are traditionally 8 axioms to check whether a set $V$ together with a field $F$ constitute a vector space. A common list of axioms can be found here. Missing from the list, however, is a ...
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1answer
67 views

What does it mean to axiomatize a logic?

I'm sorry if this question is not clearly formulated: An axiomatization, or an axiomatic system, usually means a set of axioms (i.e. a theory). A formal theory is such a set of formulas in some formal ...
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152 views

Peano's Axioms: Mathematical Philosophy

In Peano Axioms, why is it necessary to define number and successor. Does not using them imply that we know what they mean? Or could they have just as easily been any two arbitrary terms which are not ...
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1answer
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Problem in a proof that we cannot order complex numbers

The order axioms of real numbers state 1) Either $x = y$ or $x < y$ or $x > y;$ 2) If $x < y,$ then $x+z < y+z;$ 3) If $x, y > 0,$ then $xy > 0;$ 4) If $x > y$ and $y > z,$ ...
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1answer
86 views

Is there any set theory without something like the Axiom Schema of Separation?

I appreciate any insight to this question, including suggestions for other terms to learn about first. I am self-taught with regards to set theory and not a mathematician, so my question may not be ...
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0answers
34 views

Proving the distributive law in Real numbers

Given that the distributive law holds for positive real numbers a,b,c meaning that a(b+c)=ab+ac How do we extend the proof to all real numbers?
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Contrapositive of: $(x,y \in P) \implies xy \in P$, where $P$ is the set of real numbers

One of the axioms of order for real numbers read: $$(x,y \in P) \implies xy \in P$$ where P is the set of positive real numbers. Then, the contrapositive of this statement is: $$\sim ((x,y \in P) ...
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Consistency in Related Sets of Axioms?

If I have a set of axioms A = {A1, ..., An} and if I create a set of axioms B = {A1, ..., An, Con(A)}, would it be true to say that Con(A) iff Con(B)? Is there a simple counter-example to this? More ...
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Huzita Axiom 6 - Computing the Origami Trisection of an Angle

The Galois theory proof of the improssiblity of angle trisection rests on studying the triple angle formula $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$. Ruler and compass numbers can only be ...