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

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How to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$ based on the following assumption?

Let $W=$Span$\{1, x\}$, $W_1=$Span$\{x^2, x^3\}$ and $W_2=$Span$\{1+x+x^2+x^3, 1+x+x^2-x^3\}$, how to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$? $P_3(R)=W+ W_1$ because Span$\{1, ...
1
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1answer
36 views

If commutativity of vector space is omitted, can we still use other axioms to prove the commutativity?

Here I am thinking of using $-(x+y)$ and show that it equals $-(y+x)$. $-(x+y)=-x-y$ by distributivity =$-x-y+0=...$ Here I don't know how to continue, could someone suggest?
0
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1answer
20 views

How to show that the following is satisfied for all vector space axiom?

Let $V=\{a_2x^2+a_1x+a_0|a_1, a_2, a_3\in \mathbb{R}, a_2\ne 0\}$ with operation defined by $$(a_2x^2+a_1x+a_0)+(b_2x^2+b_1x+b_0)=(a_2+b_2)x^2+(a_1+b_1)x+(a_0+b_0)$$ ...
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1answer
16 views

If $x+y=(x_1y_1, …, x_ny_n)$ and $c\cdot '\ x=x^c_1, …, x^c_n$, how to show that with these two operation $V$ is a subspace?

Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., ...
2
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1answer
62 views

Different axiomatizations of set equality

I've seen two definitions (or axioms?) of set equality: $a=b \Leftrightarrow (\forall x : x \in a \Leftrightarrow x \in b)$ $a=b \Leftrightarrow (\forall x : a \in x \Leftrightarrow b \in x)$ That ...
0
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1answer
62 views

Is it possible to create a software to find formal proofs?

Let's say I have a Hilbert style system, with a few axioms and rules of inference, and I want to find a proof for some formula $\varphi$, is it possible to create an algorithm that would find a proof ...
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0answers
17 views

Peano's 3rd axiom -explain

Peano's axioms for Natural numbers. 3rd axiom from Introduction to topology by Bert Mendelson - There is one and only one object in $\mathbb{N}$ denoted by ...
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2answers
38 views

'Multiplying' by 0 in a field, field axiom proofs

The question says: The solution set was posted and there are a few things I don't quite understand from it. For the first one, I'm not entirely sure what's happening. It appears to be using the ...
0
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1answer
40 views

How to show that the axiom for vector space hold for the following operation?

So the operation is sum defined by $f+'g=f\circ g$ (composite of functions) and usual scalar multiplication. First, for $(x+y)+z=x+(y+z)$ property, $(f+'g)+'h=(f \circ g)\circ h\ne f \circ (g\circ ...
0
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1answer
35 views

Why is the axiom for vector space not satisfied by the following equation?

Vector sum $(x_1, x_2)+'(y_1, y_2)=(x_1+2y_1, 3x_2-2y_2)$ and the usual scalar multiplication $c(x_1, x_2)=(cx_1, cx_2)$. Sure additive properties does not hold for the operation but why does the ...
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3answers
434 views

Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
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3answers
72 views

Are “formulas” in Axioms of ZFC indefinite?

There is the Separation Schema in Axioms of ZFC. Where do "formulas" in this axiom comes from?Are they indefinite and is ZFC actually something like "ZFC(X)" where X is a variable which denotes a ...
1
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1answer
46 views

Are the ordered field axioms consistent?

Today in class a student asked to the professor "Are the ordered field axioms consistent?" And my prof replied something along the lines of "Yes, as we have a model of them: $\Bbb R$, this ...
0
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1answer
21 views

Can an undefined value be said to be not an element of a defined set?

With our normal axioms of set theory is it proper to say that an undefined value is not an element of a set containing all defined values? As an example we could be asking "Is the final digit in the ...
1
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1answer
76 views

Proof that the minimal inductive set does not contain any more elements than $\{\emptyset, s(\emptyset), s(s(\emptyset)),s(s(s(\emptyset))),…\}$?

I am studying axiomatic (ZF) set theory and in particular the construction of the natural numbers from axiomatic set theory. Let me start by giving some introduction to my question. The axiom of ...
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0answers
18 views

Recover the two-sided properties from a right-sided identity axiom and a left-sided inverse axiom [duplicate]

At the end of this page: http://dogschool.tripod.com/housekeeping.html It proves the left sided axioms using the right sided axioms. It then asks you at then end to prove the two-sided axioms using a ...
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3answers
86 views

How to prove $\mathbb{N}\subseteq\mathbb{R}$ given basic field axiomatisation.

I understand that $\mathbb{R}$ can be defined axiomatically in a number of different ways, including as an extension from more basic number systems (e.g., $\mathbb{N}$, $\mathbb{Z}$, or $\mathbb{Q}$) ...
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2answers
60 views

In the structure $\langle \mathbb{Q}, < \rangle$ which of the ZF axioms hold?

In the structure $\langle \mathbb{Q}, < \rangle$ which of the following axioms hold? How about when we use the weak versions of the axioms (all $\leftrightarrow$ replaced with $\rightarrow$ )? ...
5
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1answer
37 views

Axiom of Powers

Something I'm failing to understand from Halmos "Naive Set theory" book. If $\Phi $ is a collection of subsets of a set E (that is, $\Phi$ is a subcollection of $\rho (E)$), then write First of ...
3
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1answer
62 views

How can it be seen that the Von Neumann universe $V_{\omega+\omega}$ does not model the Fraenkel axiom

Consider the Von Neumann universe $V_{\omega+\omega}$. As mentioned on the Wikipedia page on Von Neumann universes, $(V_{\omega+\omega},\in)$ is a model for $\rm Z$, but not for the Fraenkel axiom of ...
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3answers
107 views

Why do we want the Axiom of the Power Set?

I'm just learning a bit about axiomatic set theory, and I'm kind of confused as to why we need/want this axiom? Does not accepting it imply that there exists some set which doesn't have a power set? ...
6
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1answer
81 views

How can one prove the axiom of collection in ZFC without using the axiom of foundation?

Say I want to prove the axiom(s) of collection from the axiom(s) of replacement. If you have the axiom of foundation, then you can use Scott's trick to do this. But suppose I'm working in a context ...
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3answers
873 views

The smallest infinity and the axiom of choice

The short version of this question is: which (natural) axiom should be added to ZF so that the statement "$\aleph_{0}$ is the smallest infinity" becomes true? A set $A$ is called infinite if it can ...
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2answers
122 views

are there different kinds of math? [closed]

I do not mean branches such as functional analysis I mean is math we use in elementary school (which I heard uses Peano's axioms) the 'correct' math? Is there math that uses other axioms? Is ...
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3answers
106 views

$\exists x ~ \forall y ~ f(x, y) \iff \forall y() ~ \exists x~ f(x, y(x))$ Name? Proof?

I was looking at potential theorems, and this one came up: $$\bigg(\exists x ~ \forall y ~:~ f(x, y) \bigg) \iff \bigg(\forall y() ~ \exists x ~:~ f(x, y(x))\bigg)$$ (where the second $y$ is ...
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2answers
91 views

Are there axioms that imply ZFC?

The question is simple: do we know if there are non-trivial axiom collections stronger than (imply but are not implied by) $ZFC$? To clarify what I mean: Do we know a way of replacing the axioms in ...
1
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1answer
36 views

Can we assign a number to each theorem stating its complexity?

I was wondering if inside an axiomatic theory it could be possible to assign each theorem a number that indicates its complexity. Theorems with small complexity numbers would be "almost axioms"; if ...
0
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1answer
51 views

Universal Specification axiom

I am currently reading Tao's Analysis I, specifically, the section about set theory and I got stuck with one exercise which consists of deducing some basic axioms of set theory assuming the axiom of ...
6
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3answers
122 views

Is my proof of the uniqueness of $0$ non-circular?

Please try to avoid jumping directly the proof, the text before it is crucial to my question as well. I had a proof of this here, but I have come to realize that the proof is circular since I implied ...
8
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2answers
67 views

Why does probability need the Axiom of pairwise disjoint events? [duplicate]

I'm a beginning student of Probability and Statistics and I've been reading the book Elementary Probability for Applications by Rick Durret. In this book, he outlines the 4 Axioms of Probability. ...
2
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2answers
84 views

How do we introduce subtraction from these field axioms?

I am familiar with two different sets of field axioms. The first one is from "Mathematical Analysis" by Apostol. It has the first $3$ usual axioms, but the $4^{th}$ one is different: Axiom 1: ...
0
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1answer
35 views

Are axiom schema of specification and axiom of specification the same terminology?

I sometimes see books just having "axiom of specification" rather than lengthy "axiom schema of specification". Are these two the same thing?
2
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0answers
29 views

A question on Axiom XI of Veblen's paper on the axioms of geometry

Recently I have started reading Oswald Veblen's A System of Axioms for Geometry. There it is written that (see page 346), Axiom XI. If there exists an infinitude of points, there exists a certain ...
5
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2answers
117 views

Is my proof of the uniqueness of $0$ correct?

I am working my way through "Mathematical Analysis" by Apostol. What I am attempting to prove is that if there exist $q_{1}$ and $q_{2}$ such that $x + q_1 = x$ and $y+q_2=y$, then $q_1=q_2$ ...
0
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0answers
41 views

Are my proofs consistent/non-circular/correct?

I am slowly working my way trough "Mathematical Analysis" by Apostol, and I decided to try my hand at a few proofs. (I asked related questions here and here, so some of the work here might be from the ...
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1answer
84 views

How should I interpret $y-x$ in this context?

I asked a related question here. I am meticulously working my way trough Mathematical Analysis by Apostol. I am reading about axioms, specifically axiom 4: Given any 2 real numbers $x$ and $y$, ...
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1answer
44 views

What are these axioms called?

I have just begun reading "Mathematical Analysis", $2^{nd}$ edition by Apostol. In the beginning of chapter $1$ we are introduced to $9$ axioms ($+1$ later). They are the field axioms and the order ...
4
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1answer
76 views

The concept of negative numbers in the $4^{th}$ field axiom

I just started working my way trough "Mathematical Analysis", $2^{nd}$ Edition by Apostol. I am reading every detail very carefully to try to get a rigorous understanding. The $4^{th}$ axiom in that ...
5
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2answers
401 views

Programming and ZFC

Suppose I have a simple program that implements an algorithm (say depth-first search), written in a simple imperative programming language with the standard for loops, recursions, conditional ...
0
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0answers
40 views

Why does Mixture Continuity imply the Archimedean Axiom?

I am having difficulty grasping why mixture continuity implies the archimedean axiom. Mixture continuity is supposed to be a stronger statement and thus the archimedean axiom should be trivially ...
4
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2answers
127 views

A structural view to the power set axiom: Is this axiom really justifiable?

The power set axiom in set theory states that the collection of the subsets of a set is a set itself. I wonder if this is a "natural" axiom in the sense that if we consider sets as the simplest ...
2
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1answer
29 views

Why would vector space addition axiom #5 be verified in this way?

Given V = $\mathbb{R}$ with the following addition and scalar multiplication: $$x+y = x+y+7$$ $$a \cdot x=ax+7(a-1)$$ where a $\in \mathbb{R}$, while proving that V is a vector space, how come the ...
4
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2answers
169 views

Axiomatic definition of the real numbers and uncountability

There are several approaches in defining the real numbers axiomatically and suppose that we have some set of axioms $A$ which completely characterize rational numbers and which does not mention ...
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1answer
60 views

What is the importance of the axiomatization of set theory? [closed]

Well I know that de axiomatization is important to establish certain laws, etc. But what other arguments do exist?
0
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1answer
76 views

Defining an equality

When we define an equality ($=$) of things, for example of vectors in $\mathbb{R}^n$ or of sets in ZF by the Axiom of Extensionality, are there properties that we need to check in order for $=$ to be ...
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4answers
62 views

Is the axiom schema of specification sufficient for solving Russell's paradox? If so, why?

This is basically a two part question, as the title indicates. I understand why unrestricted comprehension will produce paradoxes like the Russell set, but I'm less clear on the question how the axiom ...
2
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2answers
34 views

Proving double negation from an axiomatization of classical logic

Suppose we have the following axiomatic representation of classical logic : φ → (ψ → φ) (φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)) (φ ∧ ψ) → φ (φ ∧ ψ) → ψ (φ → ψ) → ((φ → χ) → (φ → (ψ ∧ χ))) φ → (φ ∨ ψ) ψ ...
1
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1answer
61 views

On the relation of Completeness Axiom of real numbers and Well Ordering Axiom

In my abstract algebra book one of the first facts stated is the Well Ordering Principle: (*) Every non-empty set of positive integers has a smallest member. In real analysis on the other hand one ...
0
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0answers
23 views

Power Set, Theory of Linear Algebra, and Axiomatic Systems

So for homework, my History of Math professor gave us these three questions: Explain why the power set of a set S (collection of all subsets) has the same cardinality as the set [all functions from ...
4
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2answers
97 views

Concept of “eventually almost surely” as an artefact of measure-theoretic axioms?

This is a serious question despite provocative title. Ever since I found out about Cox's theorem, I got quite enthusiastic about an alternative approach to formalising probability theory and started ...