All Questions
Tagged with real-numbers axioms
17
questions with no upvoted or accepted answers
7
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0
answers
701
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Geometric basis for the real numbers
I am aware of the standard method of summoning the real numbers into existence -- by considering limits of convergent sequences of quotients.
But I never actually think of real numbers in this way. I ...
- 2,146
2
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0
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48
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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 ...
- 123
2
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68
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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 ...
- 53
2
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80
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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 ...
- 311
1
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84
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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
1
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0
answers
113
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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 ...
- 212
1
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0
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279
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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 ...
- 1,093
1
vote
1
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If $x+y=(x_1y_1, ..., x_ny_n)$ and $c\cdot '\ x=x^c_1, ..., x^c_n$, how to show that with these two operation $V$ is a subspace?
Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., x^...
- 871
0
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69
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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 ...
- 11
0
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62
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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 ...
- 1,231
0
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3
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200
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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 ...
0
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0
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38
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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]{...
- 69
0
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0
answers
96
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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. ...
- 131
0
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176
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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
0
votes
2
answers
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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