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

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Congruent figures have equal areas. Why?

When we are first introduced to the idea of congruent polygons (generally triangles) in school, we often define congruent figures as "figures which have the same shape and size". However, today, ...
46
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10answers
12k views

How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
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2answers
69 views

How to deal with equivalences in proofs?

There is a part I need clarification on regarding the use of equivalence and its symmetry. From what I understand in regards to symmetry is that: $ (p \equiv q) \equiv (q \equiv p) $. Given p and q ...
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5answers
113 views

Axioms of Euclid

The axioms of Euclid are : Things which are equal to the same thing are also equal to one another. If equals be added to equals, the wholes are equal. If equals be subtracted from equals, the ...
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2answers
19 views

Prove that addition of a constant on vector spaces is bijective

What would be a nice way to deduce from the vector space axioms that $f : V_1 \longrightarrow V_2, \, x\mapsto x+v$ with constant $v$ is bijective?
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1answer
216 views

Topology: Opens vs Neighborhoods

Disclaimer: This thread is meant informative and therefore written in Q&A style. The problems are highlighted in bold face. The axiomatization of topology can be done in various ways all of ...
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0answers
60 views

How can I make a set theory arbitrarily weak and finitely axiomatizable?

So the background is I'm trying to construct an automated theorem proving system based on first order resolution to formalize problems in mathematics and number theory. I'm familiar with the notion ...
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1answer
41 views

Axiom of extension

I am learning Set Theory from the book Naive Set Theory by Halmos as part of my course. The first chapter is on the Axiom of Extension. I understand what it is but what I don't understand is why it ...
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3answers
218 views

Axiom of extensionality in ZF - pointless?

In every textbook on ZF set theory I come across the Axiom of extensionality, which basically says that if two sets have the same elements, then those two sets are equal: $$\forall x \forall y ...
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1answer
54 views

Group Axioms Motivation

Group theory is all about symmetries. Can this be seen from the axioms defining a group? Or equivalently can the group axioms be motivated from this point of view? Of course one can look at several ...
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1answer
29 views

Ill-Founded Sets in ZFC

Let $\mathbb{N}$ be the set of all finite ordinals defined as the intersection of all ordinals including the empty set and closed under successor. Consider the following set: $S_0 = \mathbb{N}$ ...
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0answers
28 views

How to establish the formula for area of a triangle using the axioms of area?

We have the following definition: AXIOMATIC DEFINITION OF AREA We assume there exists a class of $M$ of measurable sets in the plane (i.e. subsets of the plane whose area can be defined) and a set ...
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4answers
250 views

Can a Peano Set have two or more zeros?

I repeat the Peano Axioms: Zero is a number. If a is a number, the successor of a is a number. zero is not the successor of a number. Two numbers of which the successors are equal are themselves ...
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1answer
66 views

Uniform Space: Neighborhood System [closed]

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! As the idea behind uniform spaces is to represent a ...
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1answer
448 views

Need help understanding Axiom of Extensionality

I'm attempting to learn Set Theory and I'm currently working through Halmos' Naive Set Theory. I will say that I completely understand the essence of the Axiom of Extensionality. However, where I'm ...
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3answers
320 views

Axiom of Pairing

Axiom of Pairing states that if $a,b$ are sets, $\exists$ a set $A$ such that $A=\{{a,b\}}$. My question is that why we can't use Axiom of specification to define $A=\{x|x=a \vee x=b\}$?
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1answer
34 views

A problem in transitivity of set

Let $D$ be a transitive set with the following property : $$\forall a\in D:\quad a\subseteq B\,\Longrightarrow a\in B$$ Prove that $D\subseteq B$. I think there will be needed axiom of ...
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1answer
114 views

Does every mathematical principle have a proof?

My question actually narrows down to the meaning of mathematical principle. While I'm looking for some principles, they usually have their proofs, so I thought "principle" has the same meaning as ...
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0answers
158 views

What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
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0answers
37 views

Are there any formal theories that are not axiomatic?

Are there any formal theories that are not axiomatic? Could you give me some examples?
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0answers
56 views

Simple Proofs in ZFC Set Theory

So I'll keep this real short and simple. In this document on page 23 there is a list of axioms. On page 24 there is a list of theorems that come from said axioms. I can prove them all except 3 and 5. ...
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3answers
482 views

Relationship between Category theory and Axiomatic set theory

I've recently started learning Category theory- and I have a pondering- wondering if anyone can help. Is it possible for two categories to satisfy two different set-axiom system. Namely- is it ...
2
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2answers
59 views

Is it possible to formalize areas such as image processing and computer vision?

Is it possible to formalize areas such as image processing? By formalize I mean setup axioms, then derive theorems, and reason about image processing concepts and methods formally. I would say now ...
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2answers
36 views

Let $F$ be a field and $x, y\in F$. Prove:

Use field axioms to prove: a) $(−1) · (−x) = x $ b) If $x · y = 0$ then $x = 0$ or $y = 0$ I don't understand how to approach these questions. Does the field include $1$ and $0$ as well?
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1answer
39 views

Meaning of the Axiom of regularity (foundation)

Do I understand correctly that without the Axiom of regularity (aka Axiom of foundation): $$\forall x\left(x=\varnothing\ {\Large\lor}\ \exists y\left(y \in x\ {\Large\land}\ y \cap x = ...
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1answer
66 views

Equivalent formulations of the Axiom of Replacement

Let $a$, $b$, $S$ denote sets, and let $\varphi$ be a statement in two set variables. Here are two ways of expressing the Axiom of Replacement which I have seen used: 1) If $$\text{for all } a \in S, ...
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1answer
53 views

Difference between impredicative and predicative version of separation axiom

What is the difference between an impredicative and a predicative version of the separation axiom in ZFC: $$\forall x \exists y \forall z ( z\in y \leftrightarrow (z \in x \wedge \phi (z)) $$ What ...
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2answers
60 views

How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
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2answers
89 views

Help needed with axiomatic deductions

Proof axiomatically: $\vdash \forall x (\neg (Ax \rightarrow Bx) \rightarrow (\neg Ax \rightarrow \neg Bx))$ You can use in your deduction as step the equivalence of $\varphi$ with ...
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6answers
4k views

In what sense are math axioms true?

Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers. The kid asks: why? Well, it's an axiom. It's called commutativity (which is not even true for most groups). How do I ...
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5answers
399 views

What is $\mathbb{R}$, $exactly$? [closed]

$$\text{DEFINE ALL TERMS}$$ I'm unaware of any question of this kind here, and so I ask, what is $\mathbb{R}$, exactly? By exactly I mean starting from the very beginning, axiomatically and set ...
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2answers
140 views

Axiom of Regularity - Transitive set

I just managed to confuse myself completely while studying for Set Theory. We have the Axiom of regularity: $$\forall S (S\not= \emptyset \rightarrow (\exists x\in S)(S\cap x=\emptyset))$$ Now a set ...
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1answer
43 views

Well-Ordering Theorem without Axiom of Regularity

Personally, I am not fond of the Axiom of Regularity. Some alternative models in set theory use the negation of the Axiom of Regularity as an axiom (non well-founded theories). I am curious if the ...
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2answers
60 views

Can a set of theorem be derived from different sets of incompatible axioms? [closed]

Let's take T = {set of statements} that can be derived from S1 = {set of axiom}. Assuming that we keep the same derivation rules, are there any other S, with S1 & S always false, from which we can ...
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3answers
128 views

Prove the distributive law $a(b+c)=ab+ac$ for real numbers?

I have always taken these kinds of things for granted. Well of course $a(b+c)=ab+ac$! But why? The thought randomly popped in my head, and I realized that I could not prove it. Perhaps we should take ...
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2answers
197 views

When the mathematical community consider the inclusion of a new axiom?.

At first I was thinking about the axiom of choice, but let's keep it general. What motivates the inclusion of new axioms (or change the ones we already have in an already defined axiomatic theory?. It ...
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0answers
35 views

Set of true statements generated by set of axioms with a binary operator

I wondered about this and I am having a hard time formulating it as a question at all, but I hope I can express something if my wondering here. Assume we have a set $V = \{\mathbb N, +, =, X ,(,)\}$. ...
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0answers
61 views

Hilbert projection theorem without countable choice

All the proofs of the Hilbert projection theorem, existence part, that I have seen so far use countable choice (usually implicitly). Is this necessary? It seems like you might be able to leverage the ...
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0answers
45 views

Strength of “Every finite dimensional subspace of a vector space has a complement”

Does the following choice principle have a name? Every finite dimensional subspace of a vector space has a complement. Equivalently, every line inside a vector space has a complementary ...
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2answers
42 views

Problem understanding the Axiom of Foundation

I am just beginning to learn the ZF axioms of set theory, and I am having trouble understanding the Axiom of Foundation. What exactly does it mean that "every non-empty set x contains a member y such ...
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1answer
59 views

Proof using addition and multiplication axioms

I'm working on addition and multiplication axioms of integers for discrete math. I'm trying to prove (k - m) + (m - n) = k - n. The first step I took was this ...
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1answer
118 views

Is it possible to prove the axiom of infinity from the real number axioms?

What I mean by the title is whether, given that there is a class $\Bbb R$ and operations $+,\cdot,<$ that satisfy the ordered field axioms and the least upper bound axiom, you can prove the ...
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2answers
64 views

Axiom of unrestricted comprehension

I'm doing some research on naive set theory and was a little confused over the statement of the axiom of unrestricted comprehension, $\exists$B$\forall$x(x$\in$B$\iff$$\phi$(x)). I am curious as to ...
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3answers
306 views

Derive by modus ponens $[A\rightarrow(B\rightarrow C)]\rightarrow[(A\rightarrow B)\rightarrow(A\rightarrow C)]$

How could I derive by modus ponens the formula $$[A\rightarrow(B\rightarrow C)]\rightarrow[(A\rightarrow B)\rightarrow(A\rightarrow C)]$$ from, and just from, the following axiom schemata? $(A\lor ...
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3answers
56 views

Examples of when $0\cdot x\neq 0$ or $x+0\neq x$.

I'm trying to prove that the arithmetic axioms are independent by constructing a model in which all bar one of the axioms are satisfied, for each of the axioms below. A first order theory with ...
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2answers
198 views

The real numbers and the axiom of foundation

I am having a bit of confusion about the real numbers and ZF set theory (I asked a question about it a few days ago). I am a bit unsure as to why the real numbers can be in any model of ZF as they ...
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0answers
112 views

Problem solution by model theory

Sorry if that's not the right place for asking this, but didn't have anywhere else to go. I was cheking out some math problems in the Mathematical Olympiad site, and I found this one: Let $\mathbb ...
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3answers
197 views

The axiom of infinity for Zermelo–Fraenkel set theory

The axiom of infinity for Zermelo–Fraenkel set theory is stated as follows in the wikipedia page: Let $S(w)$ abbreviate $ w \cup \{w\} $, where $ w $ is some set (We can see that $\{w\}$ is a ...
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2answers
69 views

How an axiomatic system is made?

An axiom is a sentence that is taken to be true without a proof. A set of (well organised) axioms is called an axiomatic system. As consequence of these axioms we get a lot of results that we call ...
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1answer
24 views

Why is the length of one of the segments negative in the Menelaus' theorem? Aren't all distances by definition positive?

Why is the length of one of the segments negative in the Menelaus' theorem? Aren't all distances by definition positive? I think we all know the Menelaus' theorem, which claims that the following ...