All Questions
Tagged with real-numbers axioms
76
questions
0
votes
0
answers
69
views
How does Dedekind axiom imply continuity axiom
I am trying to understand a theorem that proves that the supremum axiom, Dedekind axiom, and continuity axiom are all equivalent. I have trouble understanding one point in the proof that DED implies ...
- 11
0
votes
0
answers
62
views
Proving the well ordering principle starting from the axiom of completeness. Is this topological proof valid?
While reading this SE thread, I saw in the comments someone say "the proof [that the completeness axiom implies the well ordering principle] will take some work".
However, this other thread ...
- 1,231
3
votes
3
answers
250
views
Why can we prove facts about Euclidean geometry using coordinate method?
It's easy to show that coordinate geometry based on real number axioms satisfies the Euclidean postulates.
But how do we go the other way around?
Say we prove an arbitrary* statement about Euclidean ...
- 1,488
0
votes
3
answers
200
views
Correspondence between real numbers and points of a line
Consider this fact that we all know from school mathematics:
There is a one to one correspondence between real numbers and points of a line.
But the problem is I have never seen a rigorous proof of ...
0
votes
0
answers
38
views
Using field axioms to prove the next
how can it be proved using field axioms that
$\frac{1}{\sqrt[3]{100}}=\frac{\sqrt[3]{10}}{10}$
I have the next sketch proof:
First I applied the definition of quotient. Then I used that $1=(\sqrt[3]{...
- 69
22
votes
1
answer
1k
views
Prove all 4 axioms of "less than" are necessary (for real numbers)
One way to define an ordered field is as a field $F$ with a relation $<$ that satisfies:
For all $x,y \in F$, exactly one of $x<y$, $x=y$, $y<x$ holds.
For all $x,y,z \in F$, if $x<y$ and ...
- 145k
2
votes
1
answer
206
views
Real Numbers Cannot be Constructed: Question about Constructive Mathematics
I got into a discussion with someone stemming from the set of uncomputatble numbers and how they claimed that such numbers like $\pi$ (not uncomputable but you'll see in a second) don't exist.
I was ...
- 3,105
1
vote
0
answers
84
views
In the book by Apostol "calculus volume 1" how to prove that sum of two integers is an integer?
In Apostol's book we start by defining a set called the set of real numbers which satisfies the field and order axioms. Then we define the set of positive integers as being the subset of every ...
user907912
0
votes
3
answers
123
views
Landau Foundations of Analysis Axiom 4: Is it necessary?
Landau gives 5 axioms as the foundations for deriving the theorems in the first chapter:
Axiom 1: 1 is a natural number.
Axiom 2: If $x = y$ then $x' = y'$.
Axiom 3: 1 is not a successor to any ...
- 3
8
votes
2
answers
237
views
Is this part of axiom superfluous?
In "Analysis with an introduction to proof" (5th ed.) by Steven R. Lay, the existence of a set $\mathbb{R}$, and two binary operations $+$ and $\cdot$, satisfying 15 axioms is assumed.
The ...
- 2,681
2
votes
2
answers
257
views
Is it really important to do axiomatic study of real numbers before learning Calculus? [closed]
I am currently beginning with Calculus Volume 1 by Tom M. Apostol . It has an introduction chapter divided into 4 parts namely
Historical introduction
Basic concept of set theory
A set of axioms ...
- 191
1
vote
1
answer
173
views
How to draw Axiom of Continuity : $\exists c \in\mathbb{R} :\forall a \in A, \forall b \in B \implies a \leq c \leq b$
In Real Analysis, while we are constructing the Real Numbers Axiomatically, we (in some books) define one important Axiom, Axiom of Continuity, which goes like this :
"If $A, B\subseteq\mathbb{R}$...
- 259
1
vote
1
answer
201
views
Are these axioms of real number strict？
After comparing with some other textbooks about introductory real analysis, I find that many books' content about axioms of real numbers are not strict (at least for me, I think they are not strict).
...
- 622
1
vote
2
answers
37
views
Is it legal to define a function that gives different results for 1.0 and for 1?
In programming languages I can define such function, because in most programming languages 1.0 is not 1, because 1.0 has type "float", and 1 has type "integer". In math I don't see ...
- 121
4
votes
1
answer
105
views
Completeness Axiom of $\mathbb{R}$.
I use the following as the axiom of completeness of the reals $\mathbb{R}$:
$$\forall X,Y\in \mathcal{P}(\mathbb{R})\backslash\{\emptyset\}: (\forall x\in X\quad\forall y\in Y: x\leq y) \implies \...
- 77
0
votes
0
answers
96
views
Tarski axioms of real numbers
How does the Tarski axioms of real numbers imply that for each x,y,z ( x<y if and only if x+z < y+z ) ?
By using the 1st and 6th axioms it's easy to demonstrate that x+z<y+z implies x<y. ...
- 131
-2
votes
2
answers
260
views
For any numbers $a, b,$ and $c,$ $a + b = a + c$ if and only if $b = c$ [duplicate]
I was reading about the field of real numbers $\mathbb{R},$ and a basic question arose in my mind.
How one should prove that, for any numbers $a, b,$ and $c,$ $a + b = a + c$ if and only if $b = c?$
...
- 3,794
1
vote
2
answers
68
views
Help with a proof of a consequence from the axioms of addition and multiplication
While reading through Analysis 1 by Vladimir A. Zorich, I encountered this proof which has this 1 step I can't understand. Here is the consequence and the proof:
For every $x\in \mathbb R$ the ...
- 537
0
votes
1
answer
130
views
Construction of Real Number by Dedekind Cuts [closed]
I was studying Axiomatic Set Theory, and I have 2 questions about the construction of real numbers using Dedekind cut:
We define a real number using the Dedekind cut: $x_{\mathbb{R}} = \left \{ p \...
- 108
0
votes
2
answers
113
views
Is it possible to create the smallest real positive number by axiome?
I know that with standard math there is no "smallest positive real number". But, the same way we created Aleph Null by axiome, can we create the axiome below?
...
- 125
1
vote
0
answers
113
views
Are Real Numbers a Formal System?
I don't know a lot of mathematics but I have noticed that every branch of Mathematics has the same structure: some axioms (For example in Geometry might be Euclid's Axioms, in Probability might be ...
- 212
1
vote
1
answer
271
views
How can we show that if $|x| \le 1/n$ for all natural numbers, n, then $x = 0$?
I was thinking about how to define the real number system axiomatically, and can't find anywhere a proof that $$\left[\forall n \in \mathbb{N}\left(|x| \le \frac{1}{n}\right)\right] \Rightarrow [x = 0]...
- 13
2
votes
1
answer
281
views
Prove using the axioms that the square of any number is nonnegative
How do you prove $\forall x\in \Bbb{R}, x^2 \ge 0$ using the axioms?
My lecturer hinted you should split the cases up into $x=0$ and $x \ne 0$.
The $x=0$ case seems pretty obvious: $x^2 =x \cdot ...
- 109
0
votes
3
answers
133
views
Prove using the axioms that $x>0$ implies $-x<0$
How to prove equations that if $x>0$, then $-x<0$ using the axioms of the real numbers $\Bbb{R}$ (if $x \in \Bbb{R}$)?
My university lecturer gave this as an exercise and I am stuck on which ...
- 109
0
votes
0
answers
176
views
Question of proof of archimedean property
For every real number x there exists an integer $n$ such that $n>x$.
The book is using contradiction,
Suppose $x$ is a real number such that $n≤x$ for every $n$,that mean $x$ is the upper bound ...
user743730
1
vote
2
answers
104
views
Proof that $\frac{1}{2} + \frac{1}{2} = 1$ using just the algebraic properties of $\mathbb R$
Like the title says, can you prove rigorously that $\frac{1}{2} + \frac{1}{2} = 1$ using only the nine field properties of $\mathbb R$? I don't know if addition and multiplication are supposed to be ...
- 817
1
vote
1
answer
54
views
Why is a single nonnegative number smaller than a sum of nonnegative numbers?
I know this sounds like an incredibly dumb question, but why is a single nonnegative number smaller than a sum of nonnegative numbers in a vector? I know it's true, but I want to know why it's true. ...
- 557
1
vote
1
answer
580
views
How to derive the axiom no. 15 from the Cantor's and Archimedean axiom?
How could one substitute the (15th) axiom of completeness with Archimedean and Cantor's axiom?
We discussed Cantor's axiom as well as Archimedean in analysis lectures and were told this question might ...
- 4,670
0
votes
2
answers
123
views
How can I prove this statement without using reduction to absurdity?
$\forall a,b\in\mathbb R[\forall c\in \mathbb R(c>a\implies c>b)\implies a\ge b]$
- 178
1
vote
0
answers
279
views
Are there an axioms for cartesian space?
Do we have any axioms that allow us represent Cartesian coordinates on a graph in euclidean space or is it purely intuitive? It's easy to intuitively justify where $(0,1)$ and $(1,1)$ would lie in ...
- 1,093