hello everyone a curious question for all
HOW CAN WE BELIEVE HERONS FORMULA WITHOUT PROOF
ALSO PROVE
1+1 =2 WHERE U NEED TO USE ONLY POSTULATES
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Yes. A typical way in mathematics of proving a statement can't be proven is to find a model in which it's false.
Take for example the statement that the square of every element in a group is equal to the identity element: for all x,x2=1.x,x2=1. You can't prove it since it's false in the group having three elements, and you can't prove its negation since it's true in the group having two elements.
This technique is used to show that each axiom in a system of axioms is independent of the rest of the axioms. One model is found in which it's false, but another in which it's true, but in both models all the rest of the axioms hold. See, for example, indpendence of the axiom of choice
Take for example the statement that the square of every element in a group is equal to the identity element: for all x,x2=1.x,x2=1. You can't prove it since it's false in the group having three elements, and you can't prove its negation since it's true in the group having two elements.
This technique is used to show that each axiom in a system of axioms is independent of the rest of the axioms. One model is found in which it's false, but another in which it's true, but in both models all the rest of the axioms hold. See, for example, indpendence of the axiom of choice
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