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

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1answer
41 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
406 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 $A=\{x|x=a \vee x=b\}$?
6
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1answer
268 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 ...
6
votes
1answer
141 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 ...
3
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0answers
49 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?
2
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0answers
84 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
44 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
80 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 = ...
3
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1answer
147 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
104 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
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2answers
105 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
187 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 ...
2
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1answer
63 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
107 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 ...
2
votes
3answers
858 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
46 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
79 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 ...
6
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0answers
83 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 ...
2
votes
2answers
181 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 $y$ ...
1
vote
1answer
506 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
280 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
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2answers
100 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
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3answers
57 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
67 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
101 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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0answers
146 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
72 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?
14
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2answers
329 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 ...
5
votes
1answer
361 views

How can I prove that there is no set containing itself without using axiom of foundation?

I've already found some similar questions in here (and other sites), but in most of the case, the use of axiom of foundation is required to complete the proof. Is there any way to prove $\not\exists ...
10
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0answers
125 views

Axiomatizing topology through continuous maps

Suppose we have some topological space $X$ and we somehow forgot about the topology. A friend of ours knows the topology and offers to tell us for any map $X\to Y$ into any topological space $Y$ ...
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3answers
111 views

Question regarding axiom of unions

By axiom of union for any set A there is a set B such that x belongs to B if and only if x belongs to some z which belongs to A. According to this everything is a set.My question is what would union ...
4
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1answer
72 views

Equivalence of Axiom of Regularity

So Axiom of regularity states: every non-empty set A contains an element that is disjoint from A I'm wondering if this is equivalent as any set is not a member of itself? If so, how do we prove it? ...
2
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2answers
72 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 ...
2
votes
1answer
80 views

How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
6
votes
4answers
979 views

which axiom(s) are behind the Pythagorean Theorem

There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the ...
3
votes
1answer
98 views

What is the weakest notion of “set” that we need, so that we can say the Yoneda lemma implies something about sets?

We set up a set theory axiomatically by fixing certain statements about "$\in$". There are many different set theories. The Yoneda lemma (the theorem about functors to Set) is a early but central ...
5
votes
3answers
140 views

Why can mathematicians pick and choose axioms?

Obviously, there need to be some 'self-evident' truths that we can't prove, but on which we base certain theorems; e.g. the axiom of choice lead to the well-ordering theorem (which could well be an ...
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votes
2answers
572 views

Difference between postulates, axioms, and theorems?

I'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the ...
2
votes
4answers
85 views

Are different constructions of an algebraic structure always isomorphic?

Any two complete ordered fields are isomorphic (as proved, e.g., in Spivak's Calculus; see also this question). While I understand this proof, I cannot yet appreciate why it is necessary. Given any ...
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2answers
110 views

Does $x\cdot 0 = 0$ follow from the field axioms alone?

From the field axioms alone, does it follow that $x \cdot 0 = 0$ for all $x$? All I would like is a statement that it can or cannot be done (hints not necessary). I would like to do it myself; I ...
2
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1answer
67 views

Alternative Axiomatic Systems

At least as I understand the motivation behind rigorous definitions of the foundations of mathematics (the only contender with which I'm familiar being ZFC and extensions), the idea of an axiomatic ...
3
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1answer
105 views

How do we decide which axioms are necessary?

I am studying the axioms for a complete ordered field. I have looked at different sources, some of which differ slightly in their listings. Given some construction of the reals (e.g. Dedekind cuts or ...
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0answers
86 views

Transitive property of equality and the fundamental nature of algrebra

The fundamental nature of algebra rests on the basic rule that whenever two numbers, variables, or expressions are equal, either one can be replaced at any time by the other one. For example, if we ...
3
votes
3answers
292 views

Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
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vote
5answers
137 views

Prove that $\sqrt{2}>1$

I need to prove that $\sqrt{2}>1$, but the initial assumption I am given is that $\sqrt{2}>0$. I have $\sqrt{2}>0$ so $2>0$ (multiply by $\sqrt{2}$ on each side). I don't know what my ...
2
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1answer
81 views

Is Lambda calculus a purely equational theory?

In a previous question I have been told that lambda calculs is pure syntax. I see that Lambda calculus is introduced inductively, but I don't see from what axioms it follows that: $$(\lambda x.x) M ...
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2answers
611 views

Gödel's (in)completeness theorems and the axiomatization of Euclidean geometry

In David Hilbert's 1899 Grundlagen der Geometrie, Hilbert gives a rigorous axiomatization of Euclidean geometry. As I understand it, some of Hilbert's axioms must be expressed in second order logic ...
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1answer
91 views

Closed under an operation?

So the question is write down a definition for it means to be closed under an operation. I said that a set is closed under an operation if that operation returns an element in the set when evaluated ...
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1answer
126 views

Axiom of Infinity, existence of empty set circular?

I am reading Jech's "Set Theory". First, he states that the existence of the empty set follows from the axiom of infinity. The empty set is defined, using the separation schema as $$\emptyset = \{ u ...
0
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1answer
65 views

Basic Field Properties: multiplication

I am struggling with the proofs: a) $(a^{-1})^{-1} = a$ b) $(-a)^{-1} = -a^{-1}$ I have done the rest of the theorem but it is just these two that are difficult. To prove them you can only use the ...