# [FOM] a little prose

Harvey Friedman friedman at math.ohio-state.edu
Wed May 14 18:07:18 EDT 2008

```I am writing the paper "Finite to infinite" that contains a full
treatment of the results I posted in

http://www.cs.nyu.edu/pipermail/fom/2008-April/012800.html
http://www.cs.nyu.edu/pipermail/fom/2008-April/012802.html

and more. Here is some of the prose. As indicated in the
Methodological Thesis, the prose is of limited size.

I have more prose in mind, but no corresponding rigorous
systemization. However, I like this additional prose, but I like it
only because I expect to provide further rigorous systemizations.

So I wait till the prose now will be subsumed later.

It could be the case that I thought some extended prose was truly
important - yet I had no idea how to construct and/or develop the
corresponding rigorous systemization.

In that case, under my methodology, I would eventually write such
extended prose - with profuse apologies and exhortations that people
need to find corresponding rigorous systemizations.

Excerpt from "Finite to infinite".

FINITE TO INFINITE
by
Harvey M. Friedman
May 14, 2008

We document our approach to the following "critical question".

IN WHAT SENSE IS INFINITE SET THEORY AN EXTRAPOLATION OF FINITE SET
THEORY VIA THE ADDITION OF THE AXIOM OF INFINITY?

This critical question naturally arises when teaching axiomatic set
theory. It is commonly remarked that all of the ZFC axioms except
Infinity and Foundation obviously hold in the finite sets, and that
Foundation also holds if sets are constructed by x union {y}, starting
with emptyset.

Our approach to the critical question is as follows.

i. Identify "simple" comprehension statements and schemes, in various
senses.

ii. Determine which hold in the hereditarily finite sets.

iii. Show that the ones that hold in the hereditarily finite sets,
together with the axiom of infinity, axiomatize, or nearly axiomatize,
familiar set theories such as Z (Zermelo set theory) or ZF (Zermelo
Frankel set theory).

In the course of carrying out i-iii, we have established the following.

COMPREHENSION DICHOTOMY. A "simple" comprehension axiom or axiom
scheme is either provable in standard set theory (even without
Infinity) or refutable in a very weak set theory (without Infinity). A
"simple" comprehension axiom or axiom scheme is either provable in
standard finite set theory or refutable in a very weak finite set
theory. The set of "simple" comprehension axiom or axiom schemes that
are so provable, or equivalently, so nonrefutable, or equivalently,
valid in the hereditarily finite sets, is logically equivalent to the
axioms in variants of set theory with Infinity removed.

The results of this paper apply to Extensionality, Pairing, Union,
Power Set, Separation, Replacement, and a weak form of Foundation
called "No Epsilon Cycles". These can all be obtained from "simple"
comprehension statements or schemes, in the sense of this paper. The
results do not apply to Choice or full Foundation.

Harvey Friedman
```