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

"WM" <mueckenh@rz.fh-augsburg.de> wrote in message

news:195f6374-07c4-43cf-9416-3ca841f78cd3@fe28g2000vbb.googlegroups.com...

>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)]

>http://arxiv.org/PS_cache/math/pdf/0211/0211398v1.pdf

>

>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

ZFC

as ZFC is above PA, and even higher.

1.4 The Scale Thesis: Even if you feel ZFC assumes too much or too little

(and

you do not work artificially), you will end up somewhere along this scale,

going

from PA to the large cardinals.

(What does artificial mean? For Example, there are 17 strongly

inaccessible2

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

2n

(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

primes.]

The difference between two such positions will be just where they put their

belief;

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

implication.

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

later.

6.12 Dream: Find natural properties of logics and nontrivial implications

be-

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

power

preferably concerning other parts of mathematics (see [Sh 702], possibly

specifically

derive for them).

6.18 Dream: Try to formalize and really say something4 on mathematical

beauty

and depth. Of course (length of proof)/(length of theorem) is in the right

direction,

etc.

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

22100

+1)

(e.g., thinking our universe is discrete with this size).

"