(no subject)

[nibot]

1. I have `liberated' this new icon from the CERN website.

2. The sound of rain on my room's roof is quite pleasant.

3. Isn't it cheating to require a new axiom (the ``Infinity Axiom'') in order to construct the natural numbers, and still call it ``constructive'' (versus axiomatic)? The Infinity Axiom (∃A:(∅∈A)&(∀a∈A, a+∈A)) seems tailor made for the construction of the naturals.

4. vmware is amazing. I can run Windows 2000 and Linux and Plan9 simultaneously, just as if this little computer were actually three. With Mozilla and OpenOffice running on both Win2K and Linux; and with Mindterm SSH running in a Java VM; and connected to my mailboxes via IMAP, it's clear to me that the future of computers isn't with platform independence, it's with platform irrelevancy.

5. (nonetheless, it's still impossible to get any work done in Windows without somehow connecting to or emulating a Unix environment :-)

Comments

[easwaran.livejournal.com]

It is certainly cheating in a way to add a new axiom to get the set of all natural numbers. That's why, in Principia Mathematica, Bertrand Russell didn't add Infinity as an axiom, but as a hypothesis on a conditional in front of all his theorems. (The is, they all say things like "If Infinity then fundamental theorem of number theory" or something of the sort.)

But in another sense, it's only cheating if you think of it as added specifically to get the set of all natural numbers. All you really need is to guarantee that some infinite set exists. (Don't you believe that some infinite set exists?) And the standard statement of Infinity is the easiest way to get an infinite set. If you had another infinite set, you could easily construct this one with replacement and separation.