Re: How do We Know that ZF is the Axiomatization that Proves everything Provable?

On Feb 9, 9:03 am, Charlie-Boo <shymath...@xxxxxxxxx> wrote:
On Feb 8, 4:38 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On Feb 7, 10:51 am, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

On Feb 6, 6:04 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
(Define the natural numbers to be the finite ordinals, use them
to define the integers, then the rationals, then the reals in standard
ways - then go ahead and prove all the standard theorems of
analysis. What part of that seems non-trivial?

Can the proofs be carried out using only ZFC's axioms?

Using only first order logic applied to only the ZFC axioms.

So the answer is no?  (Note: I am not asking you whether your NEXT
answer will be no.)

When we say, "using only ZFC" we MEAN "using only first order logic
applied to the ZFC axioims". OF COURSE, we need not just axioms but a
logic to APPLY to the axioms. We've been through this with you
before.

We can
expedite by adding defintions also, but properly formed defintions are
always eliminable so that whatever primitive formula we prove with
defined terminology we could also prove without the defined
terminology.

I don't understand why you keep asking this question when you could
just consult an introductory textbook on set theory and one on real
analysis to see what Ullrich just mentioned carried out in detail.

(I suggested several and everybody said, "Not them." and still they
and you haven't given me a reference.)

Sorry, but it's hard to recall all the context here. You suggested
several what? You mentioned books on set theory and analysis and
someone said "'not them"? And a reference to what? To books? Would you
like me to refer you to some specific books on topology, analysis, and
abstract algebra?

But No no no.  I'm asking them what their claim is - to define it.
Funny thing is, nobody was able to answer it - until you finally said
"No, you need additional axioms."  

I didn't say "you need additional axioms" to those of ZFC to prove
virtually all of the theorems of ordinary mathematics. Please don't
put words (especially in quote marks) in my mouth.

Then above you are trying to
backtrack by saying that sometimes ZFC's axioms suffice - which says
nothing.  Sometime CBL's axioms are used, too.

No, I didn't even claim that ZFC suffices. I said that it is a claim
that ZFC suffices for proving virtually of the theorems of ordinary
mathematics and that as far as I know that claim is correct; as far as
I know ZFC suffices for virtually all (I never said 'all') of ordinary
mathematics.

So even the one answer I get has become inconsistent.

You've shown no inconsistency in what I said.

 But do see the
point - I'm just trying to tie down what the claim is.

I don't speak for others. But I've given you a quite clear statement
of the claim.

The reason is
that the claim is baseless.  ZFC has axioms for numbers and sets.
That's it.  Anything else if almost guaranteed to take other axioms -
why would ZFC's axioms apply to a different domain?

In ZFC we have the capability of capturing - in terms of the
membership relation - notions of things such as numbers, orderings,
spaces, metric spaces, topologies, algebras, graphs, geometries, etc.;
and theorems regarding these, as they are captured or represented in
ZFC are provable in ZFC.

I'm not talking about using set expressions to "represent" other
systems.  That is a silly game of changing syntax.  And notice what it
produces:

No, it's not just syntax.

And if YOU are talking about something other than that of capturing or
representing, then you're not talking about what we're talking about.
What I claim is as I stated it. It's not more than what I stated. If
you feel it's trivial, then so be it, but what I stated is not even
intended to be what YOU are talking about.

If 0 is {} and X' is {X,{X}} then the axiom that states X'+Y=(X+Y)'
translates into {X,{X}}+Y={X+Y,{X+Y}}.  A lot of good that does!  LOL

The importance of the matter is mocked there only by isolating to a
single example. What the good is overall is having one axiom system
(ZFC, for example) and language in which we can express and prove
virtually all of ordinary mathematics. If that is not of value or
interest to you, then so be it, but that doesn't make it an incorrect
claim that ZFC can express and prove virtually all of the theorems of
ordinary mathematics.

MoeBlee
.

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