Re: infinity

>> As the size of the set of finite naturals, it is also the largest
>> element of that set, and as such, it is finite.

This assumption seems to be at the root of your confusion, Tony. I'm
surprised that you can continue to entertain it since it is trivial to
prove that there are an infinite number of finite numbers: a simple
proof by contradiction that you have already seen will do.

>> If the size of the set of finite naturals exists, then so does the
>> largest element of the set, which doesn't exist.

Again, how do you justify this statement? It's not just false, it's
absurd. You offer no explanation other than hand-waving.

>> If all members of the set are finite naturals, then my proof applies
>> to each and every one of them, and shows that no one of them can
>> possibly ever have more than a finite number of predecessors.

It is true that every natural number has only a finite number of
predecessors each.

>> If a set has NO elements with a finite number of predecessors,
>> then it cannot be infinite.

I am having trouble understanding what you mean here. Each and every
element of the set of natural numbers can have only a finite number of
predecessors, but the set itself is infinite.

.

Relevant Pages

  • Re: infinity
    ... >>means the set CANNOT be infinite. ... > "every finite natural has finitely many predecessors" ... > "The set of finite naturals is a finite set" ... Because if an element has an infinite number of successors, ...
    (sci.math)
  • Re: infinity
    ... There is NO n in N for which the set of predecessors is infinite. ... But the definition of 'infinite set' does not mention predecessors. ... >> that the set of finite naturals must be finite. ... The only contradiction come from your misconceptions. ...
    (sci.math)
  • Re: infinity
    ... There is no n with an infinite number of ... >> predecessors. ... Huyah huyah Ommmmm....." ... of any infinite set of finite naturals. ...
    (sci.math)
  • Re: infinity
    ... Is there an element with 10 predecessors? ... know that any element has n successors, then you know that at least one other ... if there is some element x an infinite number ... > N are finite naturals. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... or the value of the string I am discussing. ... There is no point at which any finite natural has an infinite ... They ARE the finite naturals. ... At which point has my abstraction differed from what you're ...
    (sci.math)