4,781
edits
Changes
Jump to navigation
Jump to search
Line 34:
Line 34: + +
→Diagonalization
::::::Foxtrot you seem to be missing the essential part of the proof in that it is a proof by contaradiction. If a set is countable then it can be well ordered without using the axiom of choice. I have assumed it is countable and then created an arbritrary ordering assuming I could. I then found an number that was not in my well ordered list. Why? Because they are uncountable and you can't do this, hence a contradiction occurs and so we have proved that the numbers between [0,1] are uncountable. If you use the axiom of choice then my number would be on the list somewhere and the proof fails. [[User:DanielB|DanielB]] 22:52, 14 June 2008 (EDT)
::::::Foxtrot you seem to be missing the essential part of the proof in that it is a proof by contaradiction. If a set is countable then it can be well ordered without using the axiom of choice. I have assumed it is countable and then created an arbritrary ordering assuming I could. I then found an number that was not in my well ordered list. Why? Because they are uncountable and you can't do this, hence a contradiction occurs and so we have proved that the numbers between [0,1] are uncountable. If you use the axiom of choice then my number would be on the list somewhere and the proof fails. [[User:DanielB|DanielB]] 22:52, 14 June 2008 (EDT)
::::::::Right, I think I got hung up on the reflexive instinct "enumerate implies use of Axiom of Choice", but since the set is assumed to be countable you don't need to worry about. I settle my objections. [[User:Foxtrot|Foxtrot]] 00:26, 15 June 2008 (EDT)