Math Is Fun Forum

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

logic help

We can use a true Statement to push forward and get a new theorem or so, but how can we prove the very first of theorems? and also , if a,b,c are statements , a--> b -->c -->a, if these statements form a circle themselves , how can we determine whether they are valid?

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Numbers are the essence of the Universe

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: logic help

You never start with nothing.  In math, we start with definitions and axioms.  Definitions are just so we make sure everyone is talking about the same thing.  Axioms are what we accept as true without proof.  If we had no axioms, we couldn't do anything.  We can't even say that things such as a set exist without axioms.

Most math is built on Zermelo-Fraenkel set theory, most of the time with the Axiom of Completeness thrown in there.  From these axioms, we can construct the natural numbers, integers, rationals, reals, complex, and so on, and we can prove that they exist.  That is, if we think of numbers as sets (which just about every mathematician does, even if they don't know it).  But there is no reason why you have to, but it's proven very useful.

if a,b,c are statements , a--> b -->c -->a, if these statements form a circle themselves , how can we determine whether they are valid?

A statement is never invalid.  It is either true or false, but never invalid.  Implications can be invalid.  Lets say we have a set A = {x, y, z}.  It's the difference between:

A has an even number of elements.

and

If A contains x, then A contains w.

Your "circular" statements above mean:

If a is true, then b and c are true.
If b is true, then c and a are true.
If c is true, then a and b are true.

In other words, a, b, and c are equivalent.  As soon as you get one, you get the other 2.  And if one if false, the other two must be false.  Now the implication a -> b or b -> c or c -> a may be invalid.  But you are assuming they are valid (by what you said in my quote).

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: logic help

oh , I see ,  I recall that there was a mathematician (I dont remember his name) said that some basic theorem? or Axiom?? can't be proved.(I read this in a book about Fermat last theorem )

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Numbers are the essence of the Universe

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: logic help

That would be the Axiom of Completeness.  Godel proved that it was independent of ZF set theory.  That is, if the axioms of ZF set theory are consistent, then ZFC (with completeness added in) is consistent.  Consistent simply means that you can't reach a contradiction with the axioms.  No one has proved that ZF set theory is consistent or inconsistent, though since so many have tried (both), it is believed to be consistent.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."