All Questions
5
questions
22
votes
1
answer
1k
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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 ...
- 145k
1
vote
1
answer
271
views
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]...
- 13
0
votes
2
answers
123
views
How can I prove this statement without using reduction to absurdity?
$\forall a,b\in\mathbb R[\forall c\in \mathbb R(c>a\implies c>b)\implies a\ge b]$
- 178
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
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)...
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