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

learn more… | top users | synonyms

3
votes
2answers
34 views

origin of syntax for mathematical equations

Bear with me, I don't have any formal training in mathematics. I wonder if there is something that accounts for the syntax of mathematical equations, some deeper logic or reasons why I know that ...
1
vote
3answers
58 views

Are there in pure mathematics ensembles of number's which not divided by them self except $0$?

In pure mathematics we know well that each number divided by him self except $0$ , the question that let me confused is: Is there a proof in pure mathematics show to us that there are others ...
0
votes
0answers
20 views

Axioms for ordered fields

If i have a set of numbers, that contains every positive number, including 0 , except the negative ones, can i claim it as an ordered field? I would say no, because you can't find an $y$ for any $x$ ...
2
votes
1answer
43 views

Axioms vs Models in projective and hyperbolic geometry

I'm studying Projective Geometry. The book I'm using begins with a little bit of history of Geometry, more precisely the history of the fifth postulate, the discovery of other geometries, etc. ...
3
votes
3answers
75 views

Axiom of infinity and empty set

The axiom of infinity is formulated as $$\exists S ( \varnothing \in S \wedge (\forall x \in S) x \cup \{x\} \in S)$$ Can someone explain why the use of $\varnothing$ in the axiom of infinity makes ...
4
votes
1answer
62 views

Proving $V=V_{\sf On}$ in $\sf Z+Reg$

Define the function $$V_0=\emptyset,\qquad V_{\alpha+1}={\cal P}(V_\alpha),\qquad V_{\delta}=\bigcup_{\beta<\delta}V_\beta.$$ Since we are working in $\sf Z$ (i.e. $\sf ZF$ without the axiom of ...
0
votes
1answer
23 views

A question about a theorem derived from a given set of postulates

Currently, I am reading the book 'Godel's proof' by Ernest Nagel and James Newman, with the forward by Douglas Hofstadter. In that book, on page 15, the authors give an example of an axiomatic system ...
0
votes
1answer
40 views

By proposition 3.21 is ether acute, right or obtuse

Let A, B, C be points on a line with A ∗ B ∗ C. Let D be a point not on that line such that angle ABD is obtuse. Show that angle CBD is acute. Do NOT use the Measurement Theorem. (Hint: By ...
0
votes
1answer
16 views

Formal axiomatic system(s) which has/have--or, alternatively, hasn't/haven't--produced at least one equation or model that matches observation

Hopefully I'm not wrong to suspect that the various formal axiomatic systems, which mathematicians develop, have varying amounts of empirical support (not that I generally know such systems, except by ...
0
votes
2answers
97 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?
-2
votes
0answers
32 views

questions about the taxicab geometry

Given: distance function given by the sum of the absolute values: ̃$d(AB) = |a1 − b1| + |a2 − b2|$, (1) What does the circle with center $O = (0, 0)$ and radius $1$ look like in taxi-cab geometry? ...
0
votes
0answers
30 views

Verifying certain congruence axioms in taxicab geometry

Given: I need some help I've shown what I have so far d(A, B) = |a1 − b1| + |a2 − b2| where A = (a1, a2) and B = (b1, b2). Some people call this ...
1
vote
1answer
58 views

Is the parallelogram law a theorem or an axiom?

I'm learning about inner product spaces and I am able to prove it within an inner product space. Is this a theorem or an axiom in euclidean geometry?(note: not the geometry of Descartes)
4
votes
0answers
90 views

Why exactly is Bourbaki difficult?

I keep hearing people say that Bourbaki is difficult for most undergraduates but I still don't understand why. Surely if it starts from definitions/axioms then practically anyone should be able to ...
0
votes
1answer
5 views

Do multiclass logistic regressions obey Kolmogorov's second axiom?

Logistic regressions were taught to me using the intuition that they approximate $\mathbb{P}(Y=y|x;\theta)$. Multiclass regressions use one-vs-all classification, selecting one $y$ and classifying ...
4
votes
1answer
48 views

What *can* Euclid prove?

It is well-known that Euclid's axioms for geometry are not up to modern standards of rigor: in particular, there are a lot of times when he used "obvious" facts about the geometric objects which were ...
1
vote
2answers
173 views

Automatic theorem prover for proving simple theorems?

Is there a simple software that I could use to practice proving theorems in my course of mathematical logic? Basically what I need is ability to 1) define what axioms and laws I am allowed to use in ...
1
vote
1answer
51 views

How to establish the formula for area of a triangle using the axioms of area?

We have the following definition: AXIOMATIC DEFINITION OF AREA We assume there exists a class of $M$ of measurable sets in the plane (i.e. subsets of the plane whose area can be defined) and a set ...
0
votes
2answers
137 views

what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic?

I am trying to get my head around the Rosser-Turquette axiomatisation of Lukasiewicz n-valued logics, but cannot really follow it. Maybe if somebody can give me the axioms for 3 and 4 valued logic ...
3
votes
0answers
27 views

Proving Boolean Logic using Axioms

I am trying to prove boolean logic formulas using axioms. I have a lot of trouble deciding what to do, which axiom to use, when to use it, etc. I've asked others on proving formulas but they have all ...
0
votes
0answers
37 views

Alternative definition of a matroid?

$\mathcal M\,$ is a nonempty subset of the set of all subsets of the finite set $X$, such that: $(1)\quad\; \neg\exists U,V\in\mathcal M:U\subset V\quad$ $(2)\quad\; A\subseteq B\in\mathcal M\wedge ...
2
votes
3answers
256 views

Existence of numbers such as $\pi^{-1}$

For my non-mathematics students (this particular class are computing), I would define $\displaystyle \frac{1}{n}$ for $n\in\mathbb{N}$ as the solution of the equation $$nx=1,$$ and then ...
-7
votes
1answer
126 views

What are all kind of “metamath” good for? Can it help me here? [closed]

Those logical theories, which deals with questions that isn't really mathematics but reach mathematics more or less, often seems to be like textbooks full of definitions, plus some theorems of the ...
1
vote
1answer
38 views

Vector space and axioms

How would i go around proving a(u+v)+ u= (a+1)u+av using axioms? I started with distributivity with respect to vector addition, associativity of addition, commutativity of addition, distributivity ...
0
votes
3answers
35 views

Proving $x^2 < y^2$ by means of the Ordering Axioms [closed]

How do I prove $x^2 < y^2$, if $0 \le x < y$ with the ordering axioms? thanks!
1
vote
1answer
74 views

The fundamental axioms of mathematics

Having known about what axioms are, I want to know whether there are some "fundamental axioms of mathematics" on which every branch of mathematics depends. If yes, what are they ? Or Do we have ...
1
vote
1answer
57 views

Some problems from section 4 of Munkres

I'm right now covering Section 4 of Topology by James R. Munkres, 2nd edition, and am stuck with the following problems in the exercise set after Section 4: Problem 8(c): Show that given $a$ with ...
44
votes
4answers
9k views

Can you explain the “Axiom of choice” in simple terms?

As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. I went to Wikipedia to see what the Axiom of ...
1
vote
1answer
39 views

How do I prove this implication using axiomatic methods?

I am trying to prove that $a >0$ if and only if $\frac{1}{a} >0$, But I am having a lot of trouble doing so. So I would have to prove both sides of the implication. That is $a>0 \implies ...
0
votes
2answers
34 views

Proving that a set A cannot contain another set B which contains A?

From the ZFC axiom of regularity, which states that every non-empty set contains an element disjoint from it, we can deduce that there is no set $A$ such that $A \in A$. A proof is outlined here: ...
7
votes
6answers
1k views

How to demystify the axioms of propositional logic?

How might I go about getting some intuition on the typical axiom schemes given for propositional logic? They seem rather mysterious at first glance. For example, these are taken from: ...
1
vote
1answer
28 views

Whitehead's axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
2
votes
3answers
85 views

Russell's paradox and axiom of separation

I don't quite understand how the axiom of separation resolves Russell's paradox in an entirely satisfactory way (without relying on other axioms). I see that it removes the immediate contradiction ...
13
votes
1answer
484 views

Can we prove the existence of $A\cup B$ without the union axiom?

If $A$ and $B$ are sets, then $A\cup B$ is defined as follows: $$A\cup B:=\{x: x\in A \, \, \,\text{ or } \, \, \, x\in B\}.$$ In ZFC, $A\cup B$ exists because $\bigcup\{A,B\}$ exists by union axiom, ...
1
vote
1answer
63 views

Axiomatic Set Theory: Why do we need the “Axiom of Union”?

I've been reviewing the axioms of ZFC, and I'm trying to make sense of why the "Axiom of Union" was put in place. While the existence of the intersection (of two) sets seems to be a "Theorem" we can ...
3
votes
1answer
45 views

Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
2
votes
2answers
81 views

Module over a ring which satisfies Whitehead's axioms of projective geometry

I asked this question(Characterization of a vector space over an associative division ring). It was pointed out that the answer was negative. So I reconsidered the problem and have come up with the ...
0
votes
1answer
33 views

Whitehead's axioms of projective geometry and a vector space over a field

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
0
votes
1answer
33 views

Axiomatic proof and Boolean algebra?

I'm trying to prove that: $$(c'd') + (bc') + (a'b'c) + (ab'c) = (b' + c')(b + c + d')$$ using an axiomatic proof (i.e. using only the basic axioms and theorems of Boolean algebra).However, no matter ...
1
vote
2answers
68 views

Definability and the Separation and Replacement Axiom Schemata

I am beginning to study Set Theory, so naturally one might begin with the axioms. When reading the schemata of separation and replacement, my initial thought was, "Why would we need separation if we ...
5
votes
1answer
77 views

What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
2
votes
1answer
149 views

Is the Bourbaki treatment of Set Theory outdated?

On page 5 of the following write up, the author asks why the Bourbaki did not notice that their system of Zermelo set theory with AC was inadequate for existing mathematics. Throughout the rest of the ...
3
votes
2answers
2k views

What is the difference between an axiom and a postulate?

I here about axioms is set theory and postulates in geometry, but they seem like the same thing. Do the mean the same thing but then are used in different instances or what? Is one word more ...
-1
votes
1answer
26 views

Axiom of infinity: What is an inductive set?

This Axiom states that there exists an inductive set. But, what is the definition of an inductive set?
1
vote
1answer
31 views

Deduce supremum of Set A

Define $$Set A=\{1-\frac 1n\ | n \in\ N\}$$A. Deduce sup Ab. Use the quanitifier definition of supremum to prove your conjecture in part (a).My attempt at the solution: I believe sup A is 1?
3
votes
2answers
44 views

Logical dependence between vector space axioms

My question is not very long, but I'd like to explain where it comes from. Consider the classical definition of vector spaces: $E$ is said to be a vector space over a field $F$ when: A) E is a ...
3
votes
1answer
59 views

Are some of the Real number axioms redundant?

We are taking a course in Real Analysis with the text: "Elementary Analysis, the theory of Calculus", by Kenneth Ross. I have also been reading a little bit of Beckenbach and Bellman's book, ...
9
votes
2answers
223 views

Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
5
votes
3answers
217 views

Proof of $x*0=0$ is invalid?

My professor told me today that while my logic was good and all my steps after the assumption were correct, my argument was invalid because I was assuming what I want to prove. I don't see how. ...
2
votes
1answer
56 views

Stephen Wolfram on axiomatic systems?

I was watching a video where Wolfram was discussing the development of Mathematics, he said something along the lines of: " There is a whole universe of possible mathematics. I was curious about this ...