On Friday, June 5, 2020 at 4:47:14 AM UTC-5, Bruno Marchal wrote: > > > On 4 Jun 2020, at 22:33, Philip Thrift <[email protected] <javascript:>> > wrote: > > > Years ago I wrote about the *Zetans* > > > http://poesophicalbits.blogspot.com/2012/04/persons-without-infinities.html > > who never imagined infinities, nor found any reason to either think of > them, or invent them. > > They have a fine definition of computers and computing, and have found no > need for anything more than finite mechanism in any of their theory of > computing. > > That we came to think "infinity" plays a role in computing (or in > computing theory, or in mathematics in general) is just an aspect of our > own peculiar psychology and history, but it is not needed. > > > > That makes some sense. You can compute without axiom of infinity, and > indeed you can define what is a computer just by using the two axioms Kxy = > x, and Sxyz = xz(yz), as I have explicitly shown on this list. Similarly > you can define a computer using only elementary arithmetic, like Gödel did > implicitly and Kleene did explicitly. But to prove anything non trivial, > you need induction, and to get semantics treated mathematically, you need > actual infinity axioms, like with the notion of real numbers, etc. > > To *understand* Kxy = x …, you need an axiom of infinity at the > meta-level, and this is required by all scientist-numbers in arithmetic, so > “infinity” is more than welcome to define the notion of observer, and for > the notion of physical *laws*. > > Bruno > > > > As my old post shows, the Zetans do have their Axiom of Infinity - called the *Axiom of Zillions*.
They are happy with their finite set of numbers, but sometimes they construct bigger numbers to add to the set. But they do not think there are numbers between the bigger numbers. There are gaps. There is no potential infinity since they don't think their galaxy will last forever. ( All this follows from https://www.jstor.org/stable/2273942 ) @philipthrift > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/dbe74497-12f7-4743-bfbb-079602af43a6o%40googlegroups.com.
