For questions on axioms, mathematical statements that are accepted as being true without a mathematical proof.

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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
45 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
24 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
24 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
58 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
29 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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A problem about binary relations

Define a binary relation symbol $\gtrsim$ on the real number set $R$ satisfying: For any two real numbers $a, b$, either a $\gtrsim$ b or b $\gtrsim$ a. If both are right, we denote it as a ...
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2answers
61 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
59 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
38 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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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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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
42 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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4answers
132 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
32 views

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
63 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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24 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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1answer
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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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30 views

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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28 views

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 ...
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1answer
75 views

There is no infinite sequence $x_1 \ni x_2 \ni x_3 \ni …$

We take the "usual" axioms of Zermelo-Fraenkel set theory (axiom of extensionality, axiom of the unordered pair, axiom of the sum set, axiom of the power set, axiom of the empty set, axiom of choice ...
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1answer
26 views

Given a collection of functions $f_i$ with the same domain, how to replace with values (w/o axiom replacement)

I know from a collection of ordered pairs we can project onto the first coordinate. I'm interested if there's a way (without using the axiom of replacement) to "project" a collection of functions onto ...
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1answer
42 views

$a, b, x \in \mathbb{Q}$ with $a \neq 0$. Is the $\frac{b}{a}$ the only possible value for x in $a \cdot x = b$

I have an exercise in my last assignment for calculus which is the following: Let $a, b, x \in \mathbb{Q}$ with $a \neq 0$. Use only the field axioms and the properties which we showed in class ...
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46 views

Axiomatisability of the class of finite groups

I'm trying to prove that the class of all finite groups is not finitely axiomatizable. I was trying to do this in the same way one proves that the class of finite sets is not axiomatizable, i.e. Let ...
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2answers
85 views

Is it possible to prove reflexive, symmetric and transitive properties of equality and the transitive property of inequality?

This may be a bit of a trivial question, but can one prove the reflexive, symmetric and transitive properties of equality and the transitive property of inequality of real numbers?(and if so, how? Is ...
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41 views

Is this proof correct using just the field axioms?

I have to show, using just the field axioms for the rational numbers, that $$(x+y)\cdot (x−y)=x \cdot x − y\cdot y$$ So I have started by: Replacing $(x + y)$ with a $k$, since addition is closed ...
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1answer
35 views

Proving this axiom system is consistent.

I want to make sure I understand the correct notation and expressions for proving that an axiom system is consistent. I have the axioms Every line is a collection of points. There are at least two ...
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0answers
28 views

Axioms for substructures of an algebra.

Given an algebra $ A $ and subset $ S $ (EG. quaternion algebra H and subset $S=\{a+bi : a,b \in R\} $) What are the axioms to show check for $ S $ is a: subalgebra linear subspace (sub vector ...
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3answers
733 views

What's so special about the group axioms?

I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly arbitrary. I'm not sure what exactly it is about the ...
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1answer
117 views

Is this the easiest calculus exercise ever?

I have a series of exercises to do, but one of these seems to be really easy to do, that it even seems strange to be asked to do. The exercise is: Use only the field axioms to show that: ...
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1answer
131 views

ZFC + not-CH, can a set with cardinality between $\aleph_0$ and $2^{\aleph_0}$ be defined?

Continuum hypothesis states, there is no set with cardinality between the integers and the reals. There is a milestone result, that CH is independent from ZFC. That means, both of ZFC + CH, and ZFC + ...
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1answer
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a problem in Stein's book 'Real analysis', relate to continuum hypothesis.

The question is from chapter 2, problem 5 in Stein's book 'Real analysis': 5.There is an ordering $≺$ of $\mathbb R$ with the property that for each $y\in\mathbb R$ the set $\{x\in\mathbb R : x ≺ ...
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1answer
56 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 ...
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2answers
67 views

Is it possible to map mathematics without advanced set theory?

I wish to make a large digraph (network) linking various proofs together in mathematics from, say, the definition of a group to Galois theory. I got in my head that I wanted to do this after reading ...
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0answers
53 views

Von Neumann universe implies Foundation

Suppose we are working in $\mathsf{ZF-Foundation}$. Additionally, let us assume that $V = \bigcup\{V_{\alpha} | \alpha \in \mathsf{Ord}\}$ holds. Now I want to show that this implies Foundation. I am ...
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1answer
40 views

Reference Request: Compendium of Hilbert Systems

Is there a standard reference that contains, for some large family of logics, Hilbert Systems for those logics? I realize I can build them using, for example, the "mass production" technique outlined ...
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6answers
147 views

How do we define arc length?

In trying to write a nice proof of the derivatives of $\sin(x)$ and $\cos(x)$, I encountered a serious problem, namely that I have never seen a proper definition of the notion of arc length. Based on ...
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1answer
76 views

Associativity in category theory [closed]

In Category Theory, the composition of morphisms must be associative. What would happen if we give up this associative law? For example Lie algebras are vector spaces with non-associative binary ...
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1answer
68 views

Difference between Hilbert program and Russel & Whitehead's Principia Mathematica

May some one explain me what is difference between Hilbert program and Russel & Whitehead's Principia Mathematica? I know both of them wanted to reduce the mathematics into a set of axioms and ...
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1answer
44 views

Is the $\varphi \to \varphi$ axiom in Hilbert calculus redundant?

When I see the Hilbert calculus in logic, I sometimes notice that $T \vdash \varphi \to \varphi$ is listed as an axiom and sometimes not. Is there some reason? Could I get it somehow from the other ...
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2answers
137 views

Is there an established method for assigning a truth value $0<t<1$ to undecidable statements?

Is there an established method for assigning a truth value $0<t<1$ to undecidable statements? Classical logic assumes every proposition is either true or false, i.e., the law of excluded ...
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1answer
125 views

Proof that the finite product of nonempty sets is nonempty without axiom of choice from ZF

How do you prove that for $X_{i} \neq \emptyset$, $i \in \{1,...,n\}$ that $\prod_{i=1}^{n} X_{i} \neq \emptyset$ only using the ZF axioms but not the Axiom of Choice? I would like to see a rigorous ...
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4answers
26 views

Deriving probability of exactly one event occuring

Show that for any events A and B, the probability that exactly one of them occur is Pr(A) + Pr(B) − 2 Pr(A ∩ B). Solution: The probability that exactly one event occurs is $$ Pr ((A \cap B^c) ...
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1answer
197 views

What are disasters with Axiom of Determinacy?

It is well-known that Axiom of Choice has several consequences which might be viewed as counter-intuitive or undesirable. For example, existence of non-measurable sets or Banach-Tarski Paradox. H. ...
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1answer
20 views

Is it an axiom that the inequalities are translation-invariant or can we prove it?

I was thinking about the inequalities on the set of real numbers. To me and everyone else, it's been taught that an inequality is translation-invariant, i.e.: $x < y \implies x + c < y + c ...
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0answers
85 views

Axioms as recreational mathematics

Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the ...
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1answer
130 views

Are there logical arguments against modern $\sf ZFC$ set theory?

As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms ...
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3answers
271 views

Is the “domain of discourse” in axiomatic set theory also a “set”?

The domain of discourse is defined by Wikipedia as the "set of entities over which certain variables of interest in some formal treatment may range." However, I believe we could not call the domain of ...
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121 views

Is the anti-foundation axiom considered constructive?

In the area of theoretical computer science that I am interested in, constructive mathematics is of practical interest because it gives algorithms that can be implemented on a computer. However, ...