All Questions
22
questions
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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
2
votes
1
answer
206
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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 ...
- 3,105
0
votes
3
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123
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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 ...
- 3
1
vote
1
answer
173
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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 $A, B\subseteq\mathbb{R}$...
- 259
1
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1
answer
201
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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).
...
- 622
1
vote
2
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68
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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 ...
- 537
2
votes
1
answer
281
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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 ...
- 109
0
votes
3
answers
133
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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 ...
- 109
1
vote
1
answer
580
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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 ...
- 4,670
2
votes
2
answers
836
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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 ...
- 3,105
3
votes
2
answers
913
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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 ...
- 1,141
2
votes
0
answers
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
3
votes
1
answer
383
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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 ...
- 2,885
2
votes
0
answers
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
vote
1
answer
242
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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 ...
