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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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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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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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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