Questions tagged [axioms]
For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.
2
votes
2answers
30 views
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 ...
0
votes
1answer
46 views
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 ...
0
votes
1answer
32 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 $\...
0
votes
1answer
44 views
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 ...
-1
votes
0answers
30 views
Let V = R^2 and the addition operation and scalar operation on u = (u1,u2) and v = (v1,v2) are defined as follows:- [closed]
(u1, u2) + (v1, v2) = (0, v2)
k(u1, u2) = (ku1, u2)
verify the following axioms:-
a) u + v = v + u
b) (u + v) + w = u + (v + w)
c)(k + m)u = ku + mu
can someone help me with that problem please.
0
votes
1answer
34 views
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 ...
1
vote
1answer
48 views
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 $...
1
vote
1answer
95 views
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 ...
0
votes
0answers
34 views
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 ...
1
vote
1answer
46 views
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 ...
0
votes
0answers
32 views
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 ...
0
votes
1answer
24 views
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 ...
1
vote
1answer
62 views
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 ...
25
votes
3answers
3k views
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.
0
votes
0answers
10 views
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 $x$ ...
1
vote
0answers
22 views
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 ...
0
votes
1answer
39 views
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 ...
1
vote
2answers
36 views
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 ...
2
votes
1answer
30 views
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 ...
0
votes
2answers
36 views
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 ...
0
votes
0answers
24 views
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 ...
3
votes
1answer
62 views
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 ...
0
votes
1answer
45 views
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:
"(...
0
votes
1answer
60 views
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, ...
1
vote
0answers
40 views
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 ...
0
votes
1answer
20 views
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 ...
0
votes
2answers
32 views
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 $$θ_1 + θ_2 := (θ_1 + θ_2) ~ \mathrm{mod} ~ 2π$$
...
0
votes
3answers
48 views
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:
-1
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1answer
26 views
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 ?
0
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0answers
214 views
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 ...
1
vote
1answer
48 views
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 ...
4
votes
0answers
68 views
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 $\bigcup A$....
6
votes
2answers
232 views
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)...
1
vote
0answers
24 views
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 ...
-1
votes
1answer
64 views
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, ...
0
votes
1answer
55 views
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 $x\sim{z}\land{y\sim{z}}\implies{x\sim{y}}$. ...
1
vote
1answer
39 views
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 ...
1
vote
1answer
39 views
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 $A\models T$.
1
vote
1answer
81 views
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.
0
votes
2answers
36 views
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 ...
2
votes
1answer
43 views
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 ...
0
votes
1answer
48 views
Axiom of Specification as in Halmos' Naive set theory
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\...
0
votes
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 ...
2
votes
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 ...
2
votes
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 ...
0
votes
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 ...
2
votes
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 ...
3
votes
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 \...
0
votes
2answers
89 views
Example of a set of real numbers that is Dedekind-finite but not finite
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?
0
votes
1answer
75 views
finding structures where the singleton axiom, the axiom of extensionality and the axiom of empty set do hold
Hey friends of maths,
I am trying to find an example that fulfills the requirements mentioned above,
more precisely:
I'd appreciate your help with finding two (or better more than 200) non-...
