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LAVRANOS said:Is 1=1 atheorem?
LAVRANOS said:I asked you from a false statement to prove atheorem because you said and I quote ((In fact,you can prove anything by assuming afalse statement))
LAVRANOS said:For your information what you said is used in proving a theorem by contradiction.
LAVRANOS said:Well did you not chalenge me by calling me (beginning mathematician)
LAVRANOS said:Now coming to your answer dx: Ist of all if 1=1 is a theorem what is( for all x,x=x)?? in which if you substitute for x=1 you get 1=1.For information again this is an axiom in equality and hense not provable and hense not a THEOREEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEM I will congratulate you should you by using ANY SET OF AXIOMS WHATSOEVER PROVE 1=1 since you said and I quote :A theorem is any statement that can be logicaly deduced from a set of axioms:
LAVRANOS said:< originally posted by dx In fact,you can prove anything by assuming a false statement> Can you be so kind as to justify your nabeve doctrine by a couple of examples or example i.e assume a false statement and then prove a theorem.
Okay, if you didn't mean "naive", what in the world is a "nabeve doctrine"?LAVRANOS said:I ment above and not naive .
HallsofIvy said:Okay, if you didn't mean "naive", what in the world is a "nabeve doctrine"?
LAVRANOS said:First of all we must decide "what is proof" and give a working definition about it. Because a proof that is right for you it might be wrong for me and the opposite.
LAVRANOS said:Because when you put 1=S(0) that is mere definition of 1 and nothing else.
LAVRANOS said:Again [tex]\forall[/tex]x (x=x) it is an equality axiom that you will find nearly in every mathematical system and not only in the Peano axioms.
Also the symmetric and transitive properties are axioms concerning equality. And again you will find in any mathematical system that uses the equality predicate.
Now you claim that 1 = 1 is a theorem? Didn't you say repeatedly that "1=1 is not a THEOREEEEEEEEEEEEEEEEEEEEEEEEEEEEEEM!". Also your proof is wrong. When you say [tex] \forall{x}, x = x [/tex] you must say what x is. The correct axiom is "for all natural numbers x, x = x". Then you must show that 1 is a natural number, or take it as an axiom.LAVRANOS said:But to help you i will prove for you that 1=1 is a theorem.
We have [tex]\forall[/tex]x (x=x)
Now using the law of logic called universal elimination we can say for x=1 then 1=1.
Since we use a general axiom and a law of logic you might say that 1=1 is a theorem.
LAVRANOS said:That law of identity emanates from the natural world surrounding us because the mountain will be itself for eternity.
Classic.That law of identity emanates from the natural world surrounding us because the mountain will be itself for eternity.
LAVRANOS said:dx said and i quote:A proof is a proof: No.
matt grime said:It is simply the case that (F=>T) is T, i.e. false implies true is true.