May 31st, 2014, 03:59 AM  #1 
Newbie Joined: May 2014 From: india Posts: 3 Thanks: 0 Math Focus: Algebra  A formal proof
What is the formal way of writing a proof?Please, Help me out.

May 31st, 2014, 05:02 AM  #2  
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  Quote:
Formal proofs in the second sense require specifying a logical calculus (formal system, or formalism). Just like a program computing a certain mathematical function can be written in different programming languages resulting in different source codes, a proof of the same statement can be written in different logical calculi resulting in different syntactic objects (for example, trees of formulas or sequences of formulas). For more, you can read these Wikipedia articles: Formal proof (it deals with sequences of formulas, but that's not the only option), Hilbert system and Natural deduction.  
May 31st, 2014, 05:23 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra 
My idea of the more linguistic formal proof is:

June 7th, 2014, 10:03 AM  #4  
Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology  Quote:
You make a statement. I think for any statement in math, it can be concluded from 8(9) axioms of ZF(C)ZermeloFraenkel axioms in a finite number of steps whether it's true or not. At least Hilbert wanted it. But Godel say no. Gentzen also says something. C marks axiom of choiceit's a crucial one, you get DISASTERS in topology without it.  
June 7th, 2014, 10:32 AM  #5 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108 
Raul, what's your point? And who is C?

June 7th, 2014, 11:06 AM  #6 
Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology 
Is my 2nd sentence true? C is axiom of choice.

June 7th, 2014, 11:28 AM  #7 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  No, any rich logical calculus has statements that it can neither prove nor refute. But, as far as I know, hunting for new axioms to be able to prove interesting results has not been a problem. Existing axioms are sufficient for almost everything so far. A more interesting problem for me is making work with formal calculi convenient and practical, not requiring significantly more effort than writing and reading paper proofs.

June 7th, 2014, 11:45 AM  #8  
Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology  Quote:
 
June 7th, 2014, 11:54 AM  #9 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  
June 7th, 2014, 01:12 PM  #10 
Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology  

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