johnny5 - an edited post.

[Johnny5]

not necessarily, though it is often presumed. the author was being more correct than the average mathematician who usually presumes that when we say "we denote by FOO those things with property BAR" that it is implicit that we are saying that this uniquely characterizes them. Ie that only FOO's have the property BAR. But that is again a convention, and you hate those.

 

Not for nothing, but this made no sense.

 

Definitions of the type found in mathematics involve 'if and only if'

 

Let me think of another example.

 

 

WRONG:

Definition: A is a set if either there is at least one X, such that X is an element of A, or A is equal to the empty set.

 

RIGHT:

Definition: A is a set if and only if (either there is at least one X, such that X is an element of A, or A is equal to the empty set.)

 

Now, perhaps the definitions above lead to a contradiction which is beyond Patrick Suppes' ability to detect, but that is another issue.

 

My point has been made, and I'm of course right on this logical point.

[Johnny5]

And while I'm at it, is the axiom of choice true or false Matt?

 

Or is it one of those kind of statements whose truth value can vary?

 

Regards

[matt grime]

Nothing in mathematics is absolutely true. The axiom is true in ZFC but independent of ZF. If it's true then every vector space has a basis, but then the Banach-Tarski paradox must hold.

[matt grime]

I do not regard there as being a consistent and complete axiomatic set theory available to man.

 

 

Of course not, as Goedel proved over 80 years ago.

[matt grime]

forget the wikipedia thing - i misread you and can't decide what the point is anymore.

[Phi for All]

Enough. This thread was designed to show one member how to save time and space in his postings and has gone on now for 3 pages. Once again we have tied up a Resident expert's valuable time to solve the problems of a member who seems to relish being a problem. This is not what we're here for.