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

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2answers
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Proof of $x*0=0$ is invalid?

My professor told me today that while my logic was good and all my steps after the assumption were correct, my argument was invalid because I was assuming what I want to prove. I don't see how. ...
2
votes
2answers
78 views
+100

Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
2
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2answers
49 views

Is is possible to prove that two systems of axioms are equivalent

The question is self explanatory. Is it possible to prove that two systems of axioms in two very different branches of math are equivalent? Is there a textbook to help me? Thanks!!
0
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2answers
51 views

Equality of Real Numbers

Is the following statement provable from the axioms of $\mathbb{R}$? If $\forall \epsilon>0$, $|r-s|\leq \epsilon$, then $r=s$.
1
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1answer
61 views

set theory, Incompleteness and axiomatic systems

Is the number of theorems that can be proved (decidable) within a certain set of axioms (for instance ZFC) is finite or infinite ? in other words, are we going to fully exhaust that set of axioms ...
1
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1answer
24 views

Writing the ZF formula for the choice function (given Well-ordering)

If every set has a well order, then the axiom of choice follows: Given a well order on $\bigcup_{i \in I} A_i$ we define the choice function in this way $f(i) =$ "the first element of $\bigcup_{i \in ...
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1answer
36 views

axiom of regularity and power sets [closed]

So axiom of regularity works for any non-empty set. Is that means that a set like this {aab, +, 50, ), (, **} and all of it's subsets are actually made of empty sets? And not just sets, but power ...
2
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1answer
16 views

Uniqueness in transfinite recursion.

Theorem of transfinite recursion: Given a well-ordered set $A$ let $\varphi(g,y)$ be a ZF formula such that for every $a \in A$ and every function $g$ with domain $I_a$ (where $I_a$ is the initial ...
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0answers
42 views

Is “autonym” an autonym?

I believe this is set theory, but I don't really know set theory, so... An autonym is a word that refers to a property that the word itself happens to possess. For example "noun" is an autonym ...
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0answers
32 views

Quotient respect to an equivalence relation.

I have the natural numbers $\mathbb{N}$, in ZF. I want to construct the integers $\mathbb{Z}$ taking the quotient respect to usual the equivalence relation $R$ on $\mathbb{N}\times\mathbb{N}$ that is ...
0
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1answer
57 views

axiom of regularity and empty set

So we have axiom of regularity which says that any set (let's call it $A$) has a subset $B$ such that $A\cap B = \emptyset$. But thinking about such sets as $C = \{1, 2\}$ and $D = \{3, 4\}$ we still ...
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0answers
58 views

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, ...
1
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2answers
73 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 ...
1
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2answers
22 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?
1
vote
1answer
43 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 ...
1
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1answer
55 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 ...
1
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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: $S_0 = \mathbb{N}$ ...
1
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0answers
31 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 ...
2
votes
4answers
259 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 ...
3
votes
5answers
138 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 ...
0
votes
1answer
68 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 ...
2
votes
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 ...
3
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3answers
323 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 $A=\{x|x=a \vee x=b\}$?
5
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1answer
222 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 ...
5
votes
1answer
116 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 ...
2
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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
63 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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2answers
37 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?
3
votes
1answer
47 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 = ...
2
votes
1answer
68 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, ...
2
votes
1answer
58 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 ...
5
votes
2answers
71 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 ...
7
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2answers
159 views

Is $\mathrm{ZFC}^E$ outright inconsistent?

From $\mathrm{ZFC},$ define a new theory $\mathrm{ZFC}^E$ by adjoining a constant symbol $E$ together with axioms to the effect that: $E$ is countable and transitive $(E,\in)$ is an elementarily ...
1
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1answer
45 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 ...
3
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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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2answers
61 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 ...
2
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3answers
138 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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0answers
36 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
64 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 ...
4
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0answers
46 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
47 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 ...
1
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1answer
78 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 ...
6
votes
1answer
122 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 ...
2
votes
2answers
66 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 ...
4
votes
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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1answer
28 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 ...
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2answers
73 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 ...
2
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0answers
114 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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1answer
64 views

what key axioms are behind calculus

what key axioms make calculus correct? I know there are axioms for real numbers, are there any other important axioms behind calculus?
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2answers
268 views

Non-constructive axiom of infinity

Looking at the ZFC axioms (in the Wikipedia version), the axiom of infinity stands out because it contains an extremely specific construction. This seems rather unelegant to me, therefore I've thought ...