Re: Cantorian pseudomathematics

david petry wrote:
> Shmuel (Seymour J.) Metz wrote:
>
> > Neither "potential infinite" nor
> > "actual infinite" are Mathematical terms. If you want to use them
> > without being laughed at, provide real definitions.
>
> They are meta-mathematical terms. When we declare that all properties
> of infinitary objects come from properties of finite approximations to
> those infinitary objects, then we are talking about the potential
> infinite. If we assert that infinitary objects can have properties that
> don't correspond to properties of finite approximations to those
> objects, then we are talking about an actual infinite.

Not every mathematical object is numerical, and thus amenable to the
idea of a "finite approximation". In what way could the collection of
all non-abelian, finite groups form a "finite approximation" of an
infinite, non-abelian group (e.g., the free group on {a,b})?

I think rather one either accepts the existence of an infinite set, or
one doesn't. If one doesn't, there is no such thing as "the free group
on {a,b}" - which, I might add, is a perfectly reasonable stance to
take.

> The problem with
> the actual infinite is that it is not observable.

This is a reasonable philosophical stance; although not one I share. It
depends on what one considers "observable"; and that is a
philosophical, not a mathematical question. De gustibus non
disputandum!

> That is, we have no
> way to test (falsify) statements about the actual infinite.

This justification is the illogical part of your argument to me.

Is it possible to "observe" that there is no largest prime number? And
yet, it's easy to prove, using a finite process; thus we have falsified
the statement "the (actual) infinite set of naturals contains a largest
prime". Euclid did it; why can't we?

I always feel that the only way you will accept a statement such as
"(2^30,402,457) - 1 is prime" [1] is if I actually demonstrate that I
fail to divide it, in order, by each of 2, 3, 4, 5, 6, 7, etc. up to
2^30,402,457. This would surely be impossible in our universe; so in
your sense could not be "observed" as a computation, and therefore it
is unfalsifiable.

But there are other ways of proving that 2^30,402,457-1 is prime (or
not). Similarly, there are perfectly finite ways of drawing conclusions
about infinite sets - given that you accept their existence to start
with.

Cheers - Chas

[1] The recently discovered Mersenne prime.

.

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