Re: infinity

In article <MPG.1d878c61676d260398a1df@xxxxxxxxxxxxxxxxxxxxxxxxx>, Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> stephen@xxxxxxxxxx said:
>> We will say the number x is finite if the set { 1, 2, 3, .. x}
>> is finite. The set {1, 2, 3, .. x} is finite if it
>> has a largest element. Clearly for each x, the
>> set {1, 2, 3, ... x} has a largest element which is x.
>> So each x is finite.
> So you define the number finite if it can be the largest member of a set of
> consecutive whole numbers, and then declare all whole numbers finite because
> they can be the largest in such a set? That's entirely circular.

No. It's not. This definition for "finite" (for sets) does not depend
on whether the largest element is "finite" as a number. It merely
depends on whether the largest element exists.

That grounds the definition.

This definition for "finite" (for numbers) does depend on the definition
of finite (for sets). But since the latter definition is already
grounded, the claimed circularity does not manifest.

While I'm not wild about these definitions, they are not circular.

John Briggs
.

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