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March 29th, 2016, 10:55 PM   #1
jee
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Unhappy 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. ]

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--- 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.
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March 30th, 2016, 01:05 AM   #2
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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.
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March 30th, 2016, 03:42 PM   #3
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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.
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