Re: infinity

Daryl McCullough said:
> Tony Orlow writes:
>
> >The fact remains that NO element in ALL of the
> >set of finite naturals, not ONE, has an infinite number of predecessors, which
> >means the set CANNOT be infinite.
>
> That implication is exactly what is under contention. You are
> claiming that
>
> "every finite natural has finitely many predecessors"
> implies
> "The set of finite naturals is a finite set"
>
> Why do you believe that implication? Is it an axiom for
> you, or do you believe that it follows from some more
> basic facts?
It follows from the definition of an infinite ordered set. An infinite set has
an infinite number of elements. If those elements are ordered, then there is a
first one, then one after that, etc. Each successive element has one more
predecessor than the last. If you have an infinite number of elements, then you
have incremented the number of predecessors an infinite number of times, and
therefore have an infinite number of predecessors.
>
> Another way to phrase the issue is this:
>
> "every finite natural has finitely many (finite) predecessors"
> implies
> "every finite natural has finitely many (finite) successors"
>
> Why do you think that implication holds?
Because if an element has an infinite number of successors, then one of those
produced after an infinite number of steps from the first, has an infinite
number of predecessors. If some element has n successors, then some other
element has n predecessors. Since there are no elements with an infinite number
of predecessors, and since an infinite number of successors of any given
element means that that element is an infinite predecessor to some other
element, there cannot be an infinite number of finite successors to any finite
natural.
>
> >The set of all elements leading up to ANY n
> >in N is FINITE.
>
> Everyone agrees with that.
>
> >There is NO element which, having been generated, is a member
> >of any infinite set of finite naturals.
>
> Everyone agrees with that.
>
> >Unless aleph_0 is the largest finite plus 1, and somehow infinite, if
> >aleph_0 is the size of the set of finite naturals, then it is finite.
>
> That is what people disagree with. Once again, if A_n = { 1, 2,... n }.
> Call a set an "initial segment" if it is any set of consecutive finite
> naturals starting with 1 (an initial segment need not have a largest
> element).
>
> You are saying
>
> "forall finite n, size(A_n) = the largest element of A_n"
>
> implies
>
> "forall A such A is an initial segment,
> if A has a size then size(A) = the largest element of A"
>
> Why do you think that implication holds? Is it an axiom, or does
> it follow from some more basic facts?
It follows from the definition of "=". If A=B, knowing the value of A gives the
value of B and vice versa. That's what equality is. So if the set size and
largest element are known to be equal, and we are given the value for one of
them, then that is also the value for the other. Equality is symmetrical. It's
not a one-way implication.
>
> --
> Daryl McCullough
> Ithaca, NY
>
>

--
Smiles,

Tony
.

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