Discussion of 'Valid method of proof'.

1. Jun 11, 2008

LAVRANOS

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

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2. Jun 11, 2008

dx

In fact, you can prove anything by assuming a false statement. This is also the reason behind Godel's theorem which says that there is no theorem deducible from a set of axioms that can prove the consistency of the axioms. The fact that a particular theorem T can be deducible from the axioms cannot tell us anything about the consistency of the axioms because if the axioms were inconsistent then T could certainly be deduced from them.

3. Jun 11, 2008

LAVRANOS

< 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.

4. Jun 11, 2008

matt grime

But that is very easy - I don't know what you think is 'naive' (assuming nabeve was supposed to mean naive) about it..

Suppose that 1=-1, then (1)^2=(-1)^2, or 1=1. So from a false presumption I have deduced a true statement. Although I don't really see the point of teaching logic to beginning mathematicians, it is one of the first things that they teach in a course on propositional logic: false implies true is true.

5. Jun 11, 2008

LAVRANOS

I ment above and not naive .Is 1=1 atheorem????????????? 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)) As to how much I know logic and particularly how logic is involved in aproof I CHALENGE you here to produce a proof of any theorem and then explisitly mention the laws of logic involved apart from anything else Any way if 1=-1 then 1=1 is a theorem congratulations you have invented new maqthematics .For your information what you said is used in proving a theorem by contradiction.

6. Jun 11, 2008

LAVRANOS

dx and mattgrime I am waiting for your answer the chalenge is still there I WILL BE GLAD to compet with you in anything and everything involving mathematical proof and proof in general

7. Jun 11, 2008

matt grime

Define theorem, please. That you dismiss 1=1 as not a theorem is just one if taste. OK, it's not deserving of the title, really, but that doesn't make it less of an example.

Last edited: Jun 11, 2008
8. Jun 11, 2008

matt grime

For what it's worth, several erroneous proofs of FLT were given on the assumption that all rings of integers in number fields are UFDs.

NB - the proofs may have had other errors too, but I'm sure that one solid proof was given (modulo the false premise), even if only for some n.

Last edited: Jun 11, 2008
9. Jun 11, 2008

dx

Yes, it is a theorem.

I don't think you understand what a theorem is. A theorem is any statement that can be logically deduced from a set of axioms.
Lets say A is a theorem. Now if you assume that its negation A' is true, you can prove any statement. Since $$A \implies A + B$$,

$$A'A \implies A'(A + B) = A'A + A'B \implies B$$.

So by assuming a contradiction we have proved an arbitrary proposition B.

No, its not. In proof by contradiction, an assumption is shown to be false by deducing a proposition that you know to be false from it. In matt grime's example, a true statement was derived from a false statement.

Last edited: Jun 11, 2008
10. Jun 11, 2008

matt grime

Oh, and what's all this nonsense about a competition? There's no competition at all, nor challenge. I'm sure you're very good at maths, but there's no need to flex those muscles and to posture about it. It is simply the case that (F=>T) is T, i.e. false implies true is true.

Last edited: Jun 11, 2008
11. Jun 11, 2008

LAVRANOS

To extent the catastrophic implications of the said doctrine.( you can prove anything by assuming a false st/ment) follow the next reasoning: 1=-1 then 1=-1or(ANY THEOREM) this we can do by using a law which is called addition introduction the above is logicaly equivalent to if 1=/-1 then ANY THEOREM BUT 1=/-1(1is not equal to -1) hense by using M. ponens we get ANY THEOREM Therefor using the above reasonig we can prove any theorem at least in the field of real Nos.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: A S for the rest of your writings i will let you think them over again and i will not ask you what a logical deduction is etc etc BUT remember the chalenge is still there particularly for mattgrim.YOU must also learn ,and this is for mattgrim ,not to call your opponent names like (begining mathematician) you must always criticise of what he knows and not who he is. And to make the chalenge better let us first define what is proof a thing that should had been done a long time ago in that forum to clear alot of mistakes occuring in some proofs and get away with a lot of stupid arguments.

12. Jun 11, 2008

LAVRANOS

Well did you not chalenge me by calling me (beginning mathematician) The chalenge is still there unless you apology .Who said i am agood mathematician,but i happen to know afew things about logic that i do not throw arround to cause impressions to other people that are ignorant of Also you must realise that titles do not imply straight thinking.In my life i have come across professors and doctors so stupid that made me ready to vomet many times .In ancient Athens every body used to go to a place called the MARKED and there in the open discuss everythimg known in the planet then.That is where civilazation was born.For your information the GREAT Aristoteles never earned any titles for hemself but he was known for what he wrote and said the following is the heart of the two valued logic that we use in maths and our ordinary life ((ει γαρ αληθες ειπειν οτι λευκον η οτι ου λευκον εστιν αναγκη ειναι λευκον η ου λευκον,και ει εστιν λευκον η ου λευκον,αληθες ην φαναι η αποφαναι.Και ει μη υπαρχει,ψευδεται,και η ψευδεται,ουχ υπαρχει.ωστε αναγκη η την καταφασιν η την αποφασιν αληθη ειναι η ψευδη)) Αριστοτελης Περι Επμηνειας,ΙΧ.18β,1-6 Υes F impies T IS TRUE Tommorow we will talk about contradiction.

13. Jun 12, 2008

matt grime

No, I didn't. I said I don't particularly like the teaching of logic to beginning mathematicians; I was thinking of a particular course I have dealt with for first year undergraduates where precisely this idea, (F=>T is T) causes some head scratching. I'm sorry if you took that aside as a personal slight.

I don't see your point: you admit (F=>T) is T, which is all that either of the two people you seem to wish to "challenge" have said. That you don't like calling the statement 1=1 a theorem is just one of taste. It was not chosen because it was all one can prove, but for the fact it was undeniably true. Do you want something more complicated? How about I prove the theorem that all finite groups have a 1-dimensional representation?

Assume all groups are abelian.

Since the sum of the squares of the simple chars of |G| equals |G|, and there are |G| conjugacy classes of an abelian group, then all reps of G are 1-dimensional, and hence at least one is 1-dimensional.

Now, why is that catastrophic? It is silly, but since the premise was false it is not important.

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14. Jun 12, 2008

matt grime

Oh, and Lame's proof of Fermat's Last theorem still remains the example I'd have in mind of this kind of thing. He assumed that rings of integers were UFDs and proved FLT. They aren't UFDs, but FLT is still a theorem.

15. Jun 12, 2008

dx

Ok, I will prove that 1 = 1 from the Peano axioms for the natural numbers. The relevant axioms are Peano's first, fifth and sixth axioms.

1. For every natural number x, x = x. That is, equality is reflexive.
5. 0 is a natural number.
6. For every natural number n, S(n) is a natural number. (S(n) is the successor of n)

It follows from (5) and (6) that 1 = S(0) is a natural number. Since 1 is a natural number, it follows from (1) that 1 = 1. Therefore, 1 = 1 is a theorem of the Peano system.

16. Jun 12, 2008

HallsofIvy

Okay, if you didn't mean "naive", what in the world is a "nabeve doctrine"?

17. Jun 12, 2008

LAVRANOS

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.
Coming now to your "proof" that 1=1 from the Peano's axioms, to me this is not a proof.
Or at least is a wrong proof.
Because when you put 1=S(0) that is mere definition of 1 and nothing else. Again $$\forall$$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.
But to help you i will prove for you that 1=1 is a theorem.
We have $$\forall$$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.
That law of identity emanates from the natural world surrounding us because the mountain will be itself for eternity.

18. Jun 12, 2008

LAVRANOS

above doctrine
Sorry for the mistake

19. Jun 12, 2008

dx

No, a proof is a proof. It can't be right for me and wrong for you (unless you disagree with standard logic). If A is a set of axioms, and if you can deduce a proposition T from A by the rules of logic, then T is a theorem of A.

Of course it is. If you don't have a definition of what 1 is how the hell can you prove a theorem about it?

Which has nothing whatsoever to do with my proof of 1 = 1. I'm not concerned with any other mathematical system other than the natural numbers. All I did was define the natural numbers by the peano postulates, and then prove that in that system 1 = 1.

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 $$\forall{x}, x = x$$ 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.

Not necessarily. Mountains can be destroyed. In the distant future, the earth will be swallowed by the sun, and the mountain will no longer be a mountain.

20. Jun 12, 2008

LAVRANOS

Yes but even destroyed will always be itself...It will be a destroyed mountain

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