Discussion of 'Valid method of proof'.

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  • #51
dx
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When he wrote "the nabeve doctrine," he meant to write, "the above doctrine."

DJ
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  • #52
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definitely you can assume the premis and arrive at a truth and then consider your theorem truth only if the way from the premises to the truth result is connected by double implications.
see 2 examples in the attached file
Lavranos,

Your statement as written is false. There are theorems whose proofs that use only single, not double implications.

For example, consider the transitivity axiom for inequality on the real number line.

It says that (for all real x)( (x<y) and (y<z) ==> x<z ). Let's call this axiom "Transitivity" for short.

Suppose we want to prove the (very simple) theorem "2 < 3." Here's a proof.

"2" and "3" are real numbers. therefore, TRANS ==> 2<3. QED.

Notice that a double inequality is not used.

The reason your statement interests me is that when I was an undergrauate, I would think about the statement you made and try to see if i could "tweak" it to make it correct. I couldn't. Let's see if I can do it now. How about:

"The shortest proof of a statement of the form "<==>" is always a proof using the double implication sign method."

I have no idea whether the above statement is true or false. I'm not even sure it is interesting.

DJ
 
  • #53
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< 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.
This question (in essence) was allegedly asked by a heckler during a talk about mathmatical logic that the great logician Bertrand Russell once gave.

Bertrand Russell had just said "Give me one false statement in mathmatics and I can prove anything you want me to."

O.K., challenged the heckler, "If 1 = 2, prove that I'M THE POPE."

(Note to moderator: That was the heckler shouting at Bertrand Russell, not me shouting at a fellow forum-ite.)

:)

With only a second's pause, Professor Russell replied,

"Well, you and the pope are two, but, if one equals two, then YOU'RE THE POPE."

:)

(Note to moderator: That was Professor Russell shouting back at the heckler. Professor Russell did not shout as loud as the heckler did.)

:)

I heard this story multiple times from multiple people when I was in graduate school at the Courant Institute of Mathematical Sceinces, N.Y.U., 1967-1969.

DJ
 
  • #54
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Deacon John i will wait until you finish the whole thread and i will answer afterwords
 
  • #55
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Read post #18...
Sorry DX. I made a "newbee" mistake. "Not suprising since I am a newbee."
DJ
 

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