Date: Feb 13, 2013 11:21 AM
Author: Scott Berg
Subject: Re: Matheology � 220

"WM" <> wrote in message

>Matheology § 220
>PA {{Peano-Arithmetik}} already tells us that the universe is
>infinite, but PA “stops” after we have all the natural numbers. {{No,
>Peano arithmetics never stops because it never reaches an end. Here
>potential and actual infinity are confused.}} ZFC goes beyond the
>natural numbers; in ZFC we can distinguish different infinite
>cardinalities such as “countable” and “uncountable”, and we can show
>that there are infinitely many cardinalities, uncountably many, etc.
>{{and we can show that there is nothing of that kind other than in
>dreams, but not in logic.}}
>[Saharon Shelah: "Logical Dreams" (2002)]
>Regards, WM

this stuff is so stale (2002), unreviewed, and you failed to post the rest:

"But there are also set theories stronger than ZFC, which are as high above
as ZFC is above PA, and even higher.
1.4 The Scale Thesis: Even if you feel ZFC assumes too much or too little
you do not work artificially), you will end up somewhere along this scale,
from PA to the large cardinals.
(What does “artificial” mean? For Example, there are 17 strongly
cardinals, the theory ZFC + “there are 84 strongly inaccessible cardinals”
is con-
tradictory and the theory ZFC + “there are 49 strongly inaccessible
cardinals” is
consistent but has no well found model.)
An extreme skeptic goes “below PA”, e.g., (s)he may doubt not only whether
(for every natural number n) necessarily exists but even whether n[log n]
exists. [In
the latter case (s)he still has a chance to prove “there are infinitely many
The difference between two such positions will be just where they put their
so the theory is quite translatable, just a matter of stress. For instance,
by one we
know that there are infinitely many primes, by the other we have an
There is a body of work supporting this, the so called equi-consistency
results (e.g.,
on real valued measurable cardinals, see later).
So far I have mainly defended accepting ZFC, as for believing in more, see
6.12 Dream: Find natural properties of logics and nontrivial implications
tween them (giving a substantial mathematical theory, of course).
6.13 Dream: Find a new logic with good model theory (like compactness, com-
pleteness theorem, interpolation and those from 6.12) and strong expressive
preferably concerning other parts of mathematics (see [Sh 702], possibly
derive for them).
6.18 Dream: Try to formalize and really say something4 on mathematical
and depth. Of course (length of proof)/(length of theorem) is in the right
6.19 Dream: Make a reasonable mathematical theory when we restrict ourselves
to the natural numbers up to n, where n is a specific natural number (say
(e.g., thinking our universe is discrete with this size).