Surprises in Logic (2016)

The final bit of Baez's article has an interesting bit here:

> So, conceivably, the concept of 'standard' natural number, and the concept of 'standard' model of Peano arithmetic, are more subjective than most mathematicians think. Perhaps some of my 'standard' natural numbers are nonstandard for you! I think most mathematicians would reject this possibility... but not all.

It's probably worth elaborating why the majority of logicians (and likely most mathematicians) believe that standard natural numbers are not subjective (although my own opinion is more mixed).

Basically the crux is, do you believe that statements of the form using all/never quantifiers such as "this machine will never halt" or "this machine will always halt" have objective truth values?

If you do, then you are implicitly subscribing to a view that the standard natural numbers objectively exist and do not depend on subjective preferences.

Natural numbers are "natural" natural numbers. That is, I would argue, that the Peano definition is the most simple, straightforward and obvious (not that I would have come up with Peano's axioms) and therefore "natural" way to define something like whole numbers. And interestingly it is the prototypical way to define an (countably) infinite set, which is the actual cause of the trouble. If you limit everything to finite sets of stuff, all the Goedel evilness goes away. But then, math would be boring. If you admit non-finite sets, you will have countably infinite sets, and those will always be isomorphic to natural numbers. I'd bet good money that if there was an alien civilization, there would be an alien named Peano.