[FOM] First, second order theories, second order characterizations

Harvey Friedman hmflogic at gmail.com
Sat Sep 7 21:27:06 EDT 2013

In http://www.cs.nyu.edu/pipermail/fom/2013-August/017552.html

I asked

1. Is CH an axiom of ZFC?
2. Is CH an "axiom" of second order ZFC"?
3. Is CH a theorem of ZFC?
4. Is CH an "axiom of second order ZFC"?

The main point of these questions is to flesh out apparent confusions I am
seeing on the FOM concerning so called  "second order systems".

On Thu, Sep 5, 2013 at 7:28 PM, Harry Deutsch <hdeutsch at ilstu.edu> wrote:

> Here's a stab at answering Harvey's question as asked.  No. Ch is not an
> axiom of ZFC.  No, it is not an axiom of ZFC second order, since this is
> not axiomatizable.  That's just part of the test, and I'm hardly an expert.
>  Harry
> On Sep 5, 2013, at 12:31 PM, Cole Leahy wrote:
> This is of course a natural  response by those who do not view "2nd order
ZFC" as having any axioms at all - in contrast to "first order ZFC".

Now look at, for example, Hewitt:

"[Dedekind 1888] and [Peano 1889] thought they had achieved success because
they had presented axioms for natural numbers and real numbers such that
models of these axioms are unique up to isomorphism with a unique
isomorphism. And later generations of mathematicians were happy to use
these axioms."

Here confusions between various notions of "axioms" and "axiom systems" and
"characterizations" are apparent, and are best fleshed out by answering

Another way of getting to the bottom of these kinds of confusions is to ask
for all full description of just what "use of these axioms" we are talking

In such cases, it is only various associated first order *formal systems*
that are in use for the mathematical reasoning, whereas various 2nd order
"characterizations" are in in the underlying mathematics.

This distinction becomes very clear as one deals with 1-4 above.

Another good exercise is to discuss the relationships between

1. First order arithmetic (PA).
2. Second order arithmetic (Z_2).
3. Second order theory of the natural numbers.

WHen reading several of the recent FOM postings, I get the impression that
the authors have not paid sufficient attention to 1-4 and 1-3.

A good test question is: is Con(ZFC) an axiom or a theorem of any of 1 - 3?

Harvey Friedman
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