My Math Forum Ruler Postulate and bijection between uncountable sets

 Real Analysis Real Analysis Math Forum

 March 29th, 2016, 10:55 PM #1 Newbie   Joined: Mar 2016 From: Sri Lanka Posts: 2 Thanks: 0 Ruler Postulate and bijection between uncountable sets why the Ruler Postulate itself is an another postulate ??? i mean there are postulates to define "Lines" "Points" and "Real Numbers" so why one cannot prove the existence of the ruler postulate using them ??? if it has to be an another postulate then there must be some building blocks behind this "ruler postulate". If it's so, what are them ? [ cause ruler postulate states that there exists a bijection between the set L (points on a line) and the real number set. ] last question --- a and b are any two different real numbers, on the real line the coordinates of points 'L' and 'R' are a and b respectively. if a < b then in the real line it's always true that the point 'L' is in the left side of the point 'R'. ( L - R manner ). so if there exist a bijection between the points on a line and the real number set, how one can be certain about the existence of that kind of unique relation ?? Please help me with these questions.
March 30th, 2016, 01:05 AM   #2
Senior Member

Joined: Jun 2015
From: England

Posts: 829
Thanks: 244

Quote:
 if it has to be an another postulate then there must be some building blocks behind this "ruler postulate". If it's so, what are them ?
Yes indeed one thing is built on another.

But

Some things are definitions

Some are postulates also called axioms

Some are propositions, also called lemmas, theorems, or results.

Definitions are also given without proof but they usually contain or discuss a single thing, mathematical object etc.

Postulates (axioms) contain something additional, they refer to relationships between defined objects. That is they contain more than one thing the object or objects and the relationship.
Again they are given without proof

Propositions are what we get when we apply the postulates to the definitions in combination.

There are usually more propositions than definitions and more definitions than axioms.

 March 30th, 2016, 03:42 PM #3 Global Moderator   Joined: May 2007 Posts: 6,526 Thanks: 588 Last question: given any bijection that preserves order, there is a related bijection that reverses the order. Examples: f(x)=x and g(x)=-x.

 Tags bijection, postulate, ruler, sets, uncountable

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post jee New Users 2 March 30th, 2016 07:19 AM mizunoami Real Analysis 1 December 6th, 2011 02:09 PM julian21 Algebra 2 September 27th, 2010 07:19 PM Bernie Applied Math 3 September 9th, 2009 01:28 AM symmetry Algebra 1 April 1st, 2007 06:11 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top