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Questions tagged [axioms]

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

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Proving the axioms [on hold]

Maths is built upon fundamental ideas (such as ‘any two points can be connected with a straight line’). These seem obviously true, but how would you even go about proving something like this?
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In Peano's Axioms are the uniqueness of the successor and $x^{\prime}=y^{\prime}\implies{x=y}$ redundant?

In Peano's Axioms are the uniqueness of the successor and the property $x^{\prime}=y^{\prime}\implies{x=y}$ redundant? This seems obvious to me, but I may be missing something. In the various forms ...
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Is it possible that $\mathcal{X} = \mathcal{Y}$, yet $\mathcal{X} \in \mathcal{Y}$? [duplicate]

Is it possible for a set to equal another set, yet the former set be an element in the latter set? I.e.: $\mathcal{X} = \mathcal{Y}$, yet $\mathcal{X} \in \mathcal{Y}$
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3answers
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Name of the basic property of equalities that if a=b then f(a)=f(b).

A basic, fundamental property of equalities is that, if one applies a function on both sides of an equality, the equality still holds. Formally: for any two objects $a$ and $b$ of type $T$ and a ...
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1answer
53 views

Is there a formalization of the link between geometry and analytical geometry?

Geometry and algebra/calculus can be formalized by axioms. Is there a global theory that combines both and establishes correspondences such as the equation of a straight line is $ax+by+c=0$, the ...
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1answer
26 views

Does the output of a total function need to be unique against the input?

I'm reading Semantics with Applications and I'm confused about their definition of a total function. They define a function like so: N : Num → Z. And then they ...
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2answers
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How are sets defined in reverse mathematics?

Currently going through Simpson's "Subsystems of second order arithmetic", which I believe is the ultimate reference in reverse mathematics, after having completed (more like peeked) Stillwell's "...
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Demonstration without the axiom of regularity [duplicate]

Show that for all set $A$ exists a set $B$ such that $B \in P(A)$ and $B \notin A$. I have tried to use the axiom schema of comprehension. I got that there is a set $B$ such that $ x\in B $ iff $x \in ...
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Was there ever an axiom rendered a theorem?

In the history of mathematics, are there notable examples of theorems which have been first considered axioms? Alternatively, was there any statement first considered an axiom that later has been ...
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1answer
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If something is provable in ZFC, can ZFC prove that ZFC proves it?

IF ZFC proves a Well-formed formula, it means, if $ZFC \vdash \phi$, then, $ZFC \vdash (ZFC \vdash \phi$) ? Generally, if an axiom system proves a wff, the axiom system can prove that it proves ...
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Axiomatize inner product spaces in terms of angles?

Inner product space is a vector space $V$ over a field $F$ together with an inner product $P:V\times V\to F$ that satisfies the inner product axioms. This inner product induces an angle $\angle (v,w) ...
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1answer
37 views

Show that the tree $T$ exists by $\Sigma_0^0$ comprehension

Currently I am reading Simpson's Subsystem of Second Order Arithmetic, Chapter IV Weak Konig Lemma. Lemma IV.1.1 (Heine/Borel theorem for $[0,1]$). The following is provable in WKL$_0.$ Given a ...
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Were there any results that were true in old axioms, but are false in modern axioms, and are not obvious paradoxes?

Recently I learned that it can't be proved that mathematical axioms are consistent. And furthermore, in 1900s math was based on an inconsistent system of axioms. So, are there any results, that were ...
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1answer
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Axiom of countable choice: a question about the domain.

From Wikipedia: The axiom of countable choice or axiom of denumerable choice, denoted $AC\omega$, is an axiom of set theory that states that every countable collection of non-empty sets must have a ...
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2answers
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Some questions about different axiomatic systems for neighbourhoods

I was thinking a few days ago about the development of topology, especially how you arrive at the concept of a topology $\tau$. I knew that a lot of initial ideas came from Hausdorff who defined a ...
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Why are the axiom of specification is an axiom schema? Why not just a single axiom?

So I was reading about the ZFC axioms, and apparently some of them are actually "axiom schemas." For example, there is the "axiom schema of specification," which basically says that give a set $A$ and ...
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1answer
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What is the point of formalization in mathematics and how does it relate to axiomatization?

Formalization is often presented (1) as a further stage after axiomatization. (It is often said that euclidian geometry was already axiomatized, but not yet formalized ( or not enough), and that the ...
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1answer
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Are axioms in mathematics comparable to hypotheses in experimental sciences? [closed]

Remark: my question deals more particularly with the axioms of set theory, arithmetic, probability theory, etc. I think the status of the axioms in geometry is clearer. The French fictitious ...
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2answers
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Vector space basics: scalar times a nonzero vector = zero implies scalar = zero?

I'm working through a linear algebra text, starting right from the axioms. So far I've understood and proved for myself that, for some vector space $V$ over a field $\mathbb{F}$: the additive ...
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1answer
46 views

Why do the hereditarily finite sets model ZF-Infinity, if we need Infinity to talk about them?

As is standard in Kunen, we can talk about $WF$ by working up to it through the "sets of rank $n$" function, $R$, defined in the usual way by iterating collection. In particular, for limit ordinals $\...
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1answer
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ZFC - Prove existence of set {A_1, A_2, A_3, …}

Good morning, I am currently taking a course in axiomatic set theory and I have encountered a problem in showing the existence of a certain infinite (countable) set. Let $A_1$, $A_2$, $A_3$, ... be ...
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1answer
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Does the Axiom schema of Replacement imply the Axiom of Infinity? [duplicate]

The axiom of infinity says that the set of natural numbers exists, while the axiom of replacement says that if an object (a member of a set) exists, then all definable mappings of that object yield ...
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1answer
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Are these two extensionality-axioms equivalent?

Let $\epsilon$ be a binary relation on a set $U$. A subset $A \subseteq U$ is called $\epsilon$-transitive iff $$a \mathrel{\epsilon} b \wedge b \in A \Rightarrow a \in A$$ for all $a,b \in U$. For $...
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1answer
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Why $x=x$? Why $x=y$ and $y=z$ imply $x=z$? (Assume, $x$, $y$ and $z$ are reals.)

I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body ...
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Finite axiomatization of second-order NBG

First-order NBG set theory is finitely axiomatizable. The proof of this basically shows that the axiom schemas, in the presence of the other axioms, can be reduced to a finite set of cases, and are ...
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1answer
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Axiom schemas vs second order axioms: which first-order predicates “exist”?

Axiom schemas, such as the PA induction schema, differ from second-order axioms in that they only hold for "definable" predicates, rather for "all" predicates. As a result, you can have non-standard ...
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Basis of Mathematics

I was reading up on Goedel's Theorems and was wondering what exactly is the basis of Mathematics today and is the discussion around an axiomatic system settled? I know that there are several attempts ...
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1answer
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Verifying axiom of substitution?

In Tao's analysis volume 1, I am introduced to this thing called the axiom of substitution. While constructing real numbers from rationals, he defined reals to be formal limits of Cauchy sequences of ...
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1answer
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Prove that the set of powers of a set, $\{ A^n : n \in \mathbb{N} \}$ exists by using ZFC axioms (without replacement).

I need to prove that the set $\{A^n : n \in \mathbb{N} \}$ using the ZFC axioms (without Replacement). My (rough) plan would be to construct some set containing "more" sets than necessary, then use ...
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Why is “points exist” not an axiom in geometry?

I am not sure why "points exist" is not an axiom in geometry, given that the other axioms are likewise primitive and seemingly as obvious.
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How to use one set of concrete axioms to build another (smaller) set of abstract axioms.

Say you have a system where your only axioms are: All foo are 10. All bar are 20. All baz are 30. Then from these axioms you can see a pattern, so you say that the underlying axiom is this: All $x$ ...
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How to determine if an axiom is consistent, independent, complete, and/or categorical

An axiomatic system is: consistent: if no logical contradiction can be derived from the axioms. (Don't see how you can prove there is no logical contradiction possible.) independent: if an axiom ...
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1answer
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How do you show that the three incidence axioms are independent of each other.

How do you show that the three incidence axioms: Incidence Axiom 1. For every pair of distinct points P and Q there is exactly one line l such that P and Q lie on l. Incidence Axiom 2. For every ...
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tautology vs axiom vs premise

I am trying to sharpen the line among the definitions of tautology, axiom and premise. What I have understood so far is this: Tautology: A statement that is proven to be true without relying on any ...
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1answer
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How to find the additive inverse and additive identity of an element of the tensor-product vector space?

The 8 axioms of holding for a vector space defined in Chapter 5 of the book Robertson's Basic Linear Algebra are easily checked for tensor product expect for existence of (unique?) additive inverse ...
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Why do we need the commutative axiom of multiplication?

I think I can prove that $ab = ba$ by other axioms. Am I wrong? Why? Edit: proof updated with notes showing where I actually used the commutative property without noticing. Proof. For any numbers ...
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Minimum eligibility to become a theory

Is there any mathematical theory without any axioms? Theories such as set theory, number theory etc., all has axioms in it. I have confusion between mathematical theory and the word theory in usage ...
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Relationship between logical axioms and tautologies?

I was reading the mathematical logic paragraph of wikipedia page on Axioms. After dividing the "types" of axioms between logical and non-logical, the article goes on like that referring to ...
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1answer
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Spivak's Calculus prologue pg 10

After introducing the properties of numbers, Spivak states: "Note, in particular, that $a>0$ if and only if $a$ is in $P$" I'm not exactly sure how to prove this. The relevant properties are: "(...
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Is there any way to prove that $x = y \Rightarrow x + z = y + z$?

Terence Tao, Analysis I, 3e, A.7 Equality (...) How equality is defined depends on the class T of objects under consideration, and to some extent is just a matter of definition. However, ...
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Incompleteness of Robinson's Arithmetic

In Smith's book about Godel's Theorems a proof is given of incompleteness of Robinson Arithmetic. It's done by producing a special interpretation of the theory: Domain is $N^* = N \cup \{a,b\}$, a ...
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1answer
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Axioms for structures with one binary operation?

In my (limited) experience of algebra, I have only seen a handful of axioms used. The wikipedia page on outline of algebraic structures lists a table of the following axioms for structures with one ...
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Confirming Axioms of Vector Spaces that rely on modular arithmetic

V is a vector space where $$V = \{\mathrm{rotations}\} = \{\theta : θ ~ \text{is a real number and} ~ 0 ≤ θ < 2π\}$$ Addition is defined by $$θ_1 + θ_2 := (θ_1 + θ_2) ~ \mathrm{mod} ~ 2π$$ ...
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How someone got the idea for the completeness axiom?

I don't understand what was the Motivation for the completeness Axiom and why Analysis and calculus would not work without it:
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Non contradiction principle

I want to know where do come exactly the contradiction principle and if a formal proof system needs it to work. Have you some books references who talks about it ?
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Euclidean incidence spaces

Structure $\left<X,B\right>$ is an $n$-dimensional Euclidean incidence space iff $B$ is ternary betweenness relation associated with $n$-dimensional Euclidean geometry. That is, Euclidean ...
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Are definitions axioms? [duplicate]

I just want to ask a very elementary question. When we introduce a "definition" in a first order logical system. For example when we say Define: $Empty(x) \iff \not \exists y (y \in x) $ Isn't ...
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When is the (great) axiom of Union really needed

Consider these two (informally stated) axioms: (Small Axiom of Union) For any two sets $A,B$ there exists the set $A \cup B$. (Great Axiom of Union) For any set $A$ there exists its union $\bigcup A$....
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Axiom checking as type checking?

There is a connection between type theory and logic, where types are propositions, and type checking performs the role of checking whether a proof of a proposition is correct (Curry-Howard isomorphism)...
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Is there a formal connection between transitivity and compositionality?

Transitivity is the property of a relation: $R_{ab}\land R_{bc} \to R_{ac}$. Compositionality is an operation on functions: $\circ : (A\to B) \times (B \to C) \to (A \to C)$. Intuitively, these are ...