# Questions tagged [axioms]

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

1,099 questions
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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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### Euclidean Geometry- Playfair's Axiom [on hold]

I'm a bit stuck on this question. Can anybody help? Prove that there exists a straight line through the origin, parallel to the line y=42, (i.e there exists a line containing the origin and not ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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?

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 ...
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### Why is it possible to use logic? [closed]

First of all: there is a philosophy of mathematics tag in this SE, so this could be the proper site for my question. I do not know how to evaluate this. When we assume an axiom to prove something, ...
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### If comparativity and reflexivity imply symmetry and transitivity how can the axioms of equivalence be orthogonal?

According to BBFSK a relation with the properties of comparativity and reflexivity satisfies symmetry and transitivity. Comparativity is defined as $x\sim{z}\land{y\sim{z}}\implies{x\sim{y}}$. ...
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### What is the difference between an axiomatization and a definition?

It is sometimes said that some things are not ever defined but instead axiomatized. What does this mean exactly and what is the difference? For example in set theory the symbol $\in$ is not ever ...
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### Prove that the class of dense linear orders cannot be axiomatized by purely universal sentenes [closed]

That is, prove that there is no set $T$ of purely universal sentences such that for every structure $A$ over the signature $\{\leq\}$, $A$ is a dense linear order iff $A\models T$.
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### Why is the “axiom of extension” an axiom? [duplicate]

I think this is a definition of = because it explains the meaning of the new symbol =. However, I wonder why the set theory thinks this as an axiom rather than a definition.
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### Are the following logical statements all axioms of propositional calculus?

I have found conflicting lists of axioms in propositional calculus in Kleene, $2002$, and on Wikipedia. From what I can tell, carefully reasoning through each of the statements reveals that are ...
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### Proving Hilbert's Axioms as Theorems in $ℝ^n$

In KG Binmore's "Topological Ideas" he says The geometric terms which appear in Hilbert's axioms are the words point, line, lie on, between and congruent. To show $\mathbb{R}^2$ is a model for ...
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As I understand, to apply the Axiom of Specification in ZF set theory, we have to specify a set $A$ to apply it to. However, I don't quite understand why Halmos says that In case $S(x)$ is $(x\... 3answers 30 views ### Is the requirement that$u+v$be in$V$if$u$and$v$are in$V$a valid axiom for the definition of a vector space? Seems to get skipped For example, at: https://en.wikipedia.org/wiki/Vector_space There are 8 axioms that a qualify a set to be a vector space. My professor also gave us 8. However, a textbook I'm reading states 10 ... 2answers 50 views ### Let$a$be a real number. If$a$is positive, then$-a$is negative. Conversely, if$a$is negative, then$-a$is positive. Let$a$be a real number. If$a$is positive, then$-a$is negative. Conversely, if$a$is negative, then$-a$is positive. Having a hard time with intuition and the obvious answer getting in the way ... 1answer 835 views ### What is the status of the Axiom of limitation of size? (adrift for almost a century now) On reviewing the wiki article Axiom of limitation of size I come away with the impression that there are issues surrounding this 'maybe too powerful' principle/heuristic/doctrine/axiom that haven't ... 3answers 58 views ### proof that weak axiom of pairing and axiom schema of specification imply axiom of pairing as the title says I am trying to give a (nearly, but not fully formal) proof that the weak axiom of pairing (i.e.$\forall x \forall y \exists p: x \in p \wedge y \in p$) together with a suitable ... 1answer 46 views ### Why is “If ψ ∈ Γ then the sequent (Γ ⊢ ψ) is correct” true? In Chiswell and Hodges Mathematical Logic the authors define a sequent as such "A sequent is an expression (Γ ⊢ ψ) (or Γ ⊢ ψ when there is no ambiguity) where ψ is a statement (the conclusion ... 1answer 106 views ### Reining in the Axiom of Power Set in ZF Given the powerset operator$\mathit P$, we have the following mapping$\tag 1 \mathcal \Phi: \mathbb N \to \Phi(\mathbb N) \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \...
Without assuming $AC$, can we find an explicit example of a subset of $\mathbb{R}$ such that it is not finite but it is Dedekind-finite?