Re: Can ZFC prove Addition is Associative?

On Feb 8, 5:08 pm, "Norman Megill" <n...@xxxxxxxxxxxxxx> wrote:
On Dec 15 2006, 10:44 pm, "Charlie-Boo" <shymath...@xxxxxxxxx> wrote:





MoeBlee wrote:
Charlie-Boo wrote:
MoeBlee wrote:
Charlie-Boo wrote:
Therefore you are going outside of ZFC.

Formal ZFC is a FIRST ORDER THEORY. The "ZFC axioms" are those that are
ADDED to "your favorite" axioms and/or inference rules for first order
logic. Anyone who has read the FIRST CHAPTER of many a basic textbook
in set theory understands that. Not you, though.

Then it sounds like you (singular or plural) agree with me. You can't
derive essentially all of mathematics from ZFC alone?

I don't want to debate what is "essentially all of mathematics".

You don't have to. The ZFC axioms don't prove anything if you have
to use "your favorite axioms". Then what good is ZFC? These other
sets of axioms and rules worked fine without ZFC. Why add ZFC?

ZFC can prove things about what sets exist [x/SE in CBL]. But when you
go outside of that question, you have to (above):

1. Do it in its own system - e.g. Number Theory in PA.
2. Express the natural numbers as set expressions.
3. Rewrite the operations as being set operations rather than
arithmetic operations.
4. Declare ZFC to axiomatize this area and sometimes even say things
like the following about 7 to 10 axioms given as being ZFC:

"There you have it, the axioms for (essentially) all of mathematics!
If you keep acopy in your wallet, you will carry with you the encoding
for all theorems ever proved and that ever will be proved, from the
most mundane to the most profound."

Do you agree with the above?

In the quotation above from my web page,http://us.metamath.org/mpegif/mmset.html#staxioms, the phrase "the
axioms" means _all_ of the preceding axioms, which include prop.
calc., pred. calc., and ZFC axioms, _not_ just the 7 preceding ZFC
axioms. In addition, there is an important note following it that
clarifies and re-emphasizes this, specifically intended to prevent
this exact kind of misunderstanding, which was apparently overlooked.
In full context, the quotation reads:

There you have it, the axioms for (essentially) all of
mathematics! ... If you keep a copy in your wallet, you will
carry with you the encoding for all theorems ever proved and that
ever will be proved, from the most mundane to the most profound.

Note. Books often make statements like "(essentially) all of
mathematics can be derived from the ZFC axioms." This should not
be taken literally-there's not much you can do with those 7 axioms
by themselves! Implicitly the authors mean the ZFC axioms plus
the axioms and rules of propositional and predicate calculus,
which total 22 axioms and 2 rules in our system. Together these
constitute what is called "ZFC set theory."

You make the claim about ZFC at several points, don't you?

C-B

I though it was clear, but I may try to rewrite it to be clearer, if
it is
still causing confusion.

--
Norm Megill ("From" address is invalid; usehttp://xri.net/=nm
instead)- Hide quoted text -

- Show quoted text -


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