All Questions
Tagged with real-numbers axioms
76
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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 ...
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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 ...
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3
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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 ...
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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 ...
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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]{...
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1
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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 ...
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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 ...
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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
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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 ...
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2
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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 ...
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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 ...
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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 $A, B\subseteq\mathbb{R}$...
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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).
...
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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 ...
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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 \...
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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. ...
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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?$
...
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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 ...
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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 \...
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2
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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?
...
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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 ...
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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]...
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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 ...
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3
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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 ...
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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
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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 ...
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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. ...
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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 ...
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123
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How can I prove this statement without using reduction to absurdity?
$\forall a,b\in\mathbb R[\forall c\in \mathbb R(c>a\implies c>b)\implies a\ge b]$
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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 ...
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How to properly define completeness of a set
A few weeks back I picked up a 1960 copy of General Theory of Functions and Integration by Taylor at a half price bookstore.
I started reading this and got up to the definition of completeness of an ...
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Confusion in least upper bound axiom
Least upper bound axiom: Every non-empty subset of $\mathbb R$ that has an upper bound must have a least upper bound.
This sounds too obvious as it works for both closed and open subsets of $\mathbb ...
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Can uncountability of reals be proved only from the axioms?
If we define real numbers, as is sometimes done, with field axioms, and order axioms, and completeness (or continuity) axiom, then, rational numbers fulfill field axioms and order axioms, but they do ...
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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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An introduction book for analysis with the axioms of Tarski
Briefly: Is there an introduction book for analysis with The axioms of Tarski as basis?
(I found them very elegant. And the most important thing for me is that the axioms are "obvious", not like the ...
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Transform a totally ordered set to a structure that is isomorphic to (R,+,.,≤)
So let $(M,\le_M)$ be a totally ordered set.
Can we define $+$ and $.$ to make $M$ isomorphic to $(\mathbb{R},+,.,\le)$?
I mean the well known axioms.
To let this possible:
$M$ is not bounded above ...
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1
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Do we need AC to have a least upper bound property?
In my analysis course, we are considering $(\mathbb{R},+,\cdot,\leq)$ as axiomatically constructed ordered field. Now, together with that, we added a completness axiom stated as follows:
Axiom: Let ...
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Abstracting Magnitude Measurement Systems (i.e. subsets of ${\mathbb R}^{\ge 0}$) via Archimedean Semirings.
I did some googling but could not find any easily accessible theory so I am going to lay out my ideas and ask if they hold water.
Definition: A PM-Semiring $M$ satisfies the following six axioms:
(1)...
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Existence of a Precise axiomatization of Eudoxus theory of magnitude
Is there a precise axiomatization of the Eudoxus theory of proportions? For example,
a) (D, +, <) is a structure such that < is a strict linear order,
b) + is an order-preserving ...
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Shouldn't there be more basic properies of real numbers in Spivak's Calculus book?
In his Calculus book, Spivak wants to establish all basic properties of real numbers so that he can prove calculus upon it. But I thought of some properties which Spivak should have also listed. And ...
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Is an equivalence relation (= sign) needed for the real number system or is a consequence of the other axioms?
My math education is based on Calculus or Real Analysis didactical books intended for bachelor's degrees, mainly read in chunks, and never went any further.
Generally, the Real Number System is said ...
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prove that if $a=b$ then $a+c=b+c$ where $a,b,c\in \mathbb R$
I was trying to prove if $l=m$ and $m=n$ then $l=n$ but when doing this I had to add $-m$ to both sides of both equations.i think it is not appropriate to proceed without proving "if $a=b$ then $a+c=b+...
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Continuity axioms and completness axioms for real numbers are the same things?
Sometime I read that Dedekind's axiom is a continuity axiom, and sometimes I read that it's a completeness axiom. Besides Dedekind's axiom is equivalent to other properties as I read here in The Main ...
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Does $\mathbb{R}$ have any axioms?
Does the set $\mathbb{R}$ of real numbers, with its usual ordering, have any axioms, or do all of its properties follow from the construction of real numbers (e.g., Dedekind cuts)?
Some analysis ...
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The axiomatic method to real number system VS the constructive method(genetic method)
According to book Georg Cantor: His Mathematics and Philosophy of the Infinite -
Joseph Warren Dauben , David Hilbert claimed that the axiomatic method to real number system is more secure than the ...
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The axiom of regularity and the real numbers
I'm having trouble understanding how there can be sufficiently many distinct elements for $\mathbb{R}$ to exist with its properties and yet still be a set.
(The ...
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Using only the field axioms of real numbers prove that $(-1)(-1) = 1$
Using only the field axioms of real numbers prove that $(-1)(-1) = 1$
(1) I start with an obvious fact:$$0 = 0$$
(2) Add $(-1)$ to both sides of the equation:
$$0 + (-1) = 0+ (-1)$$
(3) Zero is the ...
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3
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Prove, using only the field axioms of real numbers, that $0/0$ is undefined.
Prove, using only the field axioms of real numbers, that $0/0$ is undefined. I have thought about it for a while and come up with an idea how to solve this. First, I would like to prove (using field ...
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Using only the field axioms of real numbers, prove that $-x = (-1)x$ [duplicate]
Using only the field axioms of real numbers, prove that $-x = (-1)x$
Ths is how I attempted to solve this problem:
$$1+(-1)=0 \iff x(1+(-1))=0\cdot x \iff x+(-1)x=0\iff(-1)x=-x$$
However, I am not ...
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Using the axioms of real numbers prove that 0 < 1 [closed]
These are the axioms that I am allowed to use:
(1) $x + 0 = 0 + x = x$ (2)$x \cdot 1 = 1 \cdot x = x$
(3) $xy = 1 \iff y = \frac{1}{x}$, $x \neq0$
(4) $x+y = 0 \iff y = -x$
(5) $ x(y+z) = xy + xz$ ...
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