All Questions
Tagged with real-numbers axioms
76
questions
2
votes
2
answers
836
views
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
views
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
4
votes
1
answer
204
views
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 ...
- 1,802
1
vote
1
answer
149
views
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 ...
2
votes
0
answers
48
views
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
0
votes
2
answers
60
views
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 ...
- 123
3
votes
1
answer
383
views
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
3
votes
1
answer
206
views
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)...
- 11.5k
2
votes
0
answers
68
views
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
1
vote
1
answer
231
views
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 ...
- 2,104
2
votes
0
answers
80
views
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
7
votes
4
answers
11k
views
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+...
- 657
1
vote
1
answer
242
views
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 ...
1
vote
1
answer
148
views
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 ...
- 3,918
4
votes
1
answer
623
views
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 ...
- 2,267