Re: Another Inconvenient Truth

David R Tribble wrote:

Han de Bruijn wrote:

If the Axiom of Infinity comprised only _this_, I would agree with your
daughter.

David R Tribble wrote:

What else does it comprise, to the point that you don't agree
with her?

Han de Bruijn wrote:

The idea that a set can be equinumerous with a proper subset
of itself. I'm pretty sure that your six years old daughter
wouldn't know what that means.

I didn't realize that the Axiom of Infinity stated that.

The Axiom of Infinity is equivalent with that statement.
It's proven elsewhere in this thread:

http://groups.google.nl/group/sci.math/msg/a5e7198e656fad05
http://groups.google.nl/group/sci.math/msg/5899876a12c5b0c8

David R Tribble wrote:

I mean, I've always taken the A of I to mean
"there is an inductive set"
or, more informally,
"for any number k there is another number k+1".
Which is pretty close to "numbers never end".
But what else is hidden in this axiom?

Han de Bruijn wrote:

Potential infinity is pretty close to actual infinity.
But not quite.

So the difference between a potentially infinite set and
an actually infinite set is, what? Can I place the elements
that are members of one of these sets but not the other
into a separate third set, D, which is the difference
between the first two sets? If the two sets are almost,
but not quite, the same, set D should be pretty small,
yes?

There _are no_ potential infinite sets. Set theory is incompatible with
potential infinity. It needs the actual infinite, almost by definition.
(The main problem being that sets cannot be "growing") Much mathematics
can be done without set theory, though.

Han de Bruijn

.

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