Re: infinity

Tony Orlow says...

>> The only way for me to make sure that my number is bigger
>> than any finite number you can name is if my number is
>> infinite.
>Right, but why do you want a number BIGGER than all the differences
>in the set,

I want a number that is equal to or bigger than every finite
difference. But since there is no largest finite difference,
it is impossible to have a value x such that

forall differences y, y <= x

without also having

forall differences y, y < x

>> If x is a finite number, then "x+1" specifies a finite number bigger
>> than x. So if x is a finite number, it is *false* that it is
>> impossible to specify a finite number bigger than x.
>
>Right, for any GIVEN finite x, there is a y that is greater and
>finite.

>That doesn't make y infinite

We're not saying that it makes y infinite. We're saying that
if y is greater than x, then x was *not* larger than every
finite number. So we have:

If x is finite, then x+1 is a finite number greater than x.

which means

If x is finite, then it is false that x is greater than any
finite number you can name.

Logically, this has the form

A(x) implies ~B(x)

where A(x) = x is finite,
B(x) = x is greater than any finite number you can name.
Logically, A(x) implies ~B(x) is equivalent to

B(x) implies ~A(x)

That means

If x is greater than any finite number you can name,
then x is not finite.

>and it doesn't make the range infinite if for any
>given finite range you can name a larger one either.

Let's look at what we established above:

If x is greater than any finite number you can name,
then x is not finite.

Now, let x = the range of the set of all finite naturals.
Then we get

If the range is greater than any finite number you can name,
then the range is not finite.

The conclusion is that the range is not finite.

--
Daryl McCullough
Ithaca, NY

.

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