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

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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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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 ...
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79 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
147 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 $y$ ...
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1answer
289 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
276 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
94 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
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
55 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
97 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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141 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
70 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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324 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 ...
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1answer
351 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 ...
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111 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 ...
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1answer
71 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? ...
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2answers
71 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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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 ...
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4answers
902 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 ...
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1answer
96 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 ...
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3answers
137 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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2answers
366 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 ...
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4answers
83 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
109 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 ...
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1answer
49 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 ...
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1answer
92 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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56 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 ...
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3answers
284 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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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 ...
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1answer
77 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
561 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
88 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
122 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 ...
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1answer
59 views

Spivak's Axioms of a Number system. [closed]

This might be a an absurd question. I am new to studying math and have been trying not to get too caught up with symbols and formalisms. $$a + (b+c) = (a + b) + c\tag{P1}$$ $$a + 0 = a\tag{P2}$$ ...
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1answer
61 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 ...
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3answers
371 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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102 views

Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there ...
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1answer
92 views

Source request of axiom of Archimedes

I'm a little confused with axiom of Archimedes has a proof since it is an axiom. So I'm guessing there's a historical reason that this property of ordered field was given such a name. Is there any ...
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1answer
116 views

Axioms for the hyperrationals

I'm working on a comparison between a set theoretical and an axiomatic construction of the hyperrational numbers $^*\mathbb Q$. So far I have only found the construction of $^*\mathbb Q$ by using ...
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1answer
112 views

Vector Spaces of a set of continuous funtions

I am having trouble with a homework problem. This is the problem: Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify at least one of ...
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27 views

How to translate these defitions of bipartiteness to each other?

Let $(V,E)$ be a bipartite graph. I'm trying to capture this property and I've come up with two definitions and am surprised/confused that one uses a negation and the other doesn't - still I can't ...
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1answer
125 views

Can one define $\langle x,y\rangle$ in $P(C)$?

I study at course Foundations of Mathematics the below definitions and lemma: $\langle x,y\rangle:=\{\{x\},\{x,y\}\}$ (from Kuratowski 1921) $\langle x,y\rangle:=\{\{\{x\},\varnothing\}\{\{y\}\}\}$ ...
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2answers
163 views

Axiom of Extensionality - Why not called equality?

Axiom of Extensionality, I understand that if two sets have exactly the same members they are equal. However why it is called Extensionality? Why not equality?
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6answers
273 views

What is the right interpretation of the axiom of extensionality

A set $a$ can be called extensional if it has the following propery: $$\forall b\left[\forall x\left[x\in b\iff x\in a\right]\Rightarrow a=b\right]$$ Based on this the axiom of extensionality can be ...
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2answers
232 views

What is the relation between axiomatic set theory and logical quantifiers?

On the one hand, the logical predicates $\forall$ and $\exists$ are defined using the concept of a Domain of Discourse, which itself is defined as a set (at least according to wikipedia). On the ...
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3answers
107 views

Formal proof of: $x>y$ and $b>0$ implies $bx>by$?

Property: If $x,y,b \in \mathbb{R}$ and $x>y$ and $b>0$, then $bx>by$. What is a formal (low-level) proof of this result? Or is this property taken as axiomatic? The motivation for this ...
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1answer
63 views

Proving via axioms, that for given set $A$, $P(P(A))$ exists

The question itself: For a given set A, prove P(P(A)) exists. You may only use the axiom of pairing, axiom of union and axiom of empty set. This is how I solved it: Let A be the given set. ...
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1answer
167 views

A model of geometry with the negation of Pasch’s axiom?

How would a geometry with all the usual axioms of Euclidean geometry, except that instead of Pasch’s axiom, we take it negation, look like?
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2answers
72 views

Axiom of regularity and ordinal ranks

I am trying to prove that the following two statements are equivalent: Axiom of regularity $\forall x \exists \alpha (\alpha $ is an ordinal and $ x \in V_\alpha)$ I believe I understand how to ...