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

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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
79 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
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4answers
867 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
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1answer
95 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
132 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
277 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
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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
108 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
45 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
86 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
48 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
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3answers
278 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 ...
2
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1answer
76 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 ...
7
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2answers
530 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
82 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
116 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
54 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. $$a + (b+c) = (a + b) + c\tag{P1}$$ $$a + 0 = a\tag{P2}$$ ...
0
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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 ...
5
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3answers
369 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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0answers
97 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 ...
3
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1answer
89 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 ...
5
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1answer
101 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 ...
0
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1answer
108 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 ...
0
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0answers
26 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 ...
4
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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
158 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?
5
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6answers
269 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 ...
5
votes
2answers
227 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 ...
3
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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. ...
5
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1answer
162 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?
2
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2answers
62 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 ...
3
votes
1answer
103 views

When do surjections split in ZF? Two surjections imply bijection?

We have that the Axiom of Choice is equivalent to the principle that every surjection has a right inverse. However, without the Axiom of Choice we can determine for some $X$ that $X\succeq Y\implies ...
0
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2answers
100 views

How do the ZFC axioms produce the ideas of order?

The notion of order (and cardinality, for that matter) seems so basic to me that I can't imagine how it could be derived from anything. In an answer to a previous question I learned that all the ...
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3answers
181 views

Why is this considered to be an AXIOM?

One of the axioms out there is: if n and m are integers and they have the same next integer then n=m But why is it considered to be an axiom since I can easily prove it: Have the same next integer ...
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2answers
90 views

ZF Set Theory Axiom of Infinity

Could someone please state and explain the axiom of infinity in ZF set theory? This isn't homework, it's just something that has interested me for awhile.
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5answers
282 views

Please recommend a nice and concise math book on probability theory.

My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in ...
4
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3answers
627 views

Using field axioms for a simple proof

Question: If $F$ is a field, and $a, b, c \in F$, then prove that if $a+b = a+c$, then $b=c$ by using the axioms for a field. Relevant information: Field Axioms (for $a, b, c \in F$): ...
0
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1answer
118 views

Axiom of Completeness for set of integers

If $A$ is a subset of the integers $\mathbb{Z}$, and is bounded above, then A has a supremum $\alpha$ that is an element of the integers $\mathbb{Z}$. Is this statement true?
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1answer
83 views

Doubts on the axiom of union?

I'm reading Guerino Mazzolla's Comprehensive Mathematics for Computer Scientists 1: Axiom 3 (Axiom of Union) If $a$ is a set, then there is a set $$\{x| \text{ there exists an element } b \in ...
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2answers
226 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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2answers
55 views

Showing a class is a set.

The problem is to determine that a given class is or is not a set only using the basic axioms of modern set theory and some examples of classes that are or are not sets. The class in question is ...
3
votes
2answers
569 views

A set is not an element of itself.

I know that in modern set theory that for a given set $A$, $A \notin A$, specifically by the axiom of regularity. However, I'm not permitted to use this axiom in my proof. What I am permitted to use ...
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2answers
72 views

Is this a helpful way of thinking about modular arithmetic?

Could one consider the structure of $\mathbb{Z_{n}}$ under addition and multiplication to simply be that of the ordinary integers, with one additional axiom - namely that $n=0$? So we would have ...
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3answers
274 views

Why is zero important?

I am not sure whether this question is more appropriate here or in theoretical computer science. I leave it to the wisdom of moderators. On the computer science site I came across the following ...
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3answers
239 views

Axiom schema of specification (formula arguments)

Some sources define the formula like this: $$ \forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \varphi(x, w_1, \ldots, w_n , A) ] ) $$ Why ...
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3answers
72 views

One question about additive identity arising from Apostol's field axiom in the Mathematical Analysis 2nd edition

In the begining of this book, the field axiom 4 talks about "Given any two real numbers x and y, there exists a real number z such that x+z=y and this z is denoted by y-x." Therefore, for each x, we ...
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1answer
63 views

Multiple context available for the AC?

I am surprised at the language used in connection with the axiom of choice. From the answer to a question a made (which turned out to be duplicate) about involvement of AC in Wiles’ proof of Fermat’s ...
2
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2answers
104 views

Proving order in real number set

Can we prove such a statement, or is it axiomatic ? $$ \forall (x,y,z) \in \mathbb{R}^3 ∶(x \leq y)∧(y \leq z)⇒(x \leq z) $$