# If ZFC is incomplete it can not prove anything

1. Philosophy Forum
2. » Logic
3. » If ZFC is incomplete it can not prove anything

Tue 29 Apr, 2008 12:23 am
the australian philosopher colin leslie dean points out that If ZFC is
incomplete it can not prove anything
If ZFC is consistent it is incomplete i.e it has statements which cannot
be proven true or false
thus

ZFC is used to prove things in mathematics
but
ZFC can only prove this if all its statement can be proven to be true
but
ZFC has statements which cannot be proven true or false
thus it cant prove anything

ZFC being undecidable cant be used to prove anything as it has statements
which cant be proven thus without those statements being proven these
statements cant prove anything

THE ARGUMENT FOR THIS IS

the australian philosopher colin leslie dean points out that Mathematics is
systems of epistemological holism
set theory
arithmetics
geometry
algebra
etc
are systems of epistemological holisim

epistemological holism means

a systems statement coher ie dont contradict with every other statement in
the system A systems statements interlock they share a common logic and
are involved enblock in every proof.A systems statements face the
tribunal of proof as a corporate body of statements. A systems
statements about mathematics face the tribunal of proof not
individually but only as a corporate body.

thus
if a statement contradicts another statement then the system as a
corporate body enblock falls apart into inconsistency

hence skolems paradox reduces set theory thus ZFC to inconsistency

ALSO
a systems statements face the tribunal of proof as a corporate body of
statements. A systems statements about mathematics face the tribunal of
proof not individually but only as a corporate body.

thus systems which are incomplete ie there is one statement that cant be
proven then the system enblock cant prove anything

thus the systems ZFC, PA, Q due to there, incompleteness cant prove
anything

TO BE CLEAR

holism is where there is a proper fit of statements within a system.
This involves 1) the statements coher with each other ie there is logical
consistency This condition is met by mathematical systems thus they are
holistic

2)statements give mutual inferential support to each other. This
condition is met by mathematical systems thus they are holistic

as a corollary

Thus a systems statements face the tribunal of proof as a corporate
body of statements. A systems statements about mathematics face the
tribunal of proof not individually but only as a corporate body

mathematics fit an holistic system quite well

thus systems which are incomplete ie ZFC Q PA etc

Arjen

Tue 29 Apr, 2008 10:35 am
@pam69ur,
I would like to point out at first that I am not a mathematician and also no genius logician. I know a little about a lot I think. I know for one thing that logic is a framework for math. I know that what works out in logic is because of a few serious errors concerning paradoxes. This is solved by the ex falso sequitur quodlibet rule. By blindly following this rule one can ignore ontological problems which cause paradoxes because there is a "solution". This "solution" has no equivalent in math exapt a predicate letter suc as (factor) X I believe.

Is this the direction you are going?

[/stage setting]

pam69ur

Tue 29 Apr, 2008 02:44 pm
@pam69ur,
Quote:

Is this the direction you are going?

no

what i am saying is systems that are incomplete ie that have unprovable statements in them ie ZFC cannot prove anything

because

statements give mutual inferential support to each other. This
condition is met by mathematical systems thus they are holistic

as a corollary

Thus a systems statements face the tribunal of proof as a corporate
body of statements. A systems statements about mathematics face the
tribunal of proof not individually but only as a corporate body

the statements in a system are interconnected
if one statement is wrong them the system cant prove anything

also
if one statement cant be proven then likewise the system cant prove anything

de Silentio

Tue 29 Apr, 2008 06:14 pm
@pam69ur,
Quote:

logic is a framework for math

Just a question to further my learning, I thought math a framework for logic.

VideCorSpoon

Tue 29 Apr, 2008 06:53 pm
@de Silentio,
Pam69ur, since this is a topic within logic, I should say, and not to be mean, but to give a respectful opinion, you need some serious grand enaction of hypothetical syllogism. I'm sensing a lot of jumbled logic and terminology. It seems a necessity to have ordered propositional logic before quantification logic, especially when addressing Zermelo Fraenkel set theory.

pam69ur

Tue 29 Apr, 2008 07:17 pm
@pam69ur,
Quote:
t seems a necessity to have ordered propositional logic before quantification logic, especially when addressing Zermelo Fraenkel set theory.

not so
if ZFC is incomplete
propositional logic or quantification logic, are irrelevant
it follows from mathematical holism that it cant prove anything if it has unproven statements

just as if ZFC had a wrong statement
propositional logic or quantification logic, are irrelevant
as again ZFC could not prove anything

Arjen

Wed 30 Apr, 2008 05:52 am
@pam69ur,
@ De silentio: You are wrong. Math was simply the way to formulate things prior to discovery of the workings. In reality math is based on logic and perception. Therefore the logic is the framework and the a priori part of math.

@ pam69ur:
Reading your post replying to mine I cannot conclude other then that you do not understand the evidence which is delivered. What do you think people take it to mean and to which you are now rising up?

Khethil

Wed 30 Apr, 2008 06:19 am
@pam69ur,
Please forgive this wanton toss-in of a thought that might be off the thread (nuke away, powers that be, if you feel appropriate - no hard feelins).

But I've always gotten a kick out of math. As a concept/framework invented by humans initially (presumably) to describe quantities, we then invent the notion of adding them together, and multiplication (to describe more quantity) then stand back, in starry-eyed awe and say, "Wow! Such grace! Look how they all match up!".

While I can't deny the utility of the system in all its forms, this humble math-deficient dreamer penitently requests we not make it more than it is: A self-invented system that, while useful, has no more grace than any other self-asserted, self-constructed conceptualization.

Arjen

Wed 30 Apr, 2008 06:34 am
@pam69ur,
@ Khethill:
The thing about math is that not the conclusions, but the connectives are stable. We intuitively know things like +,-, etc. those are a priori to human reasoning. Therefore math is more then just a man-made system. The system we are using isn't though. There are many different maths. Binary for example.

de Silentio

Wed 30 Apr, 2008 06:54 am
@pam69ur,
Quote:

@ De silentio: You are wrong.

I thought I was wrong, but I didn't know why. Thanks for so bluntly pointing it out.

Arjen

Wed 30 Apr, 2008 08:18 am
@de Silentio,
de Silentio wrote:
I thought I was wrong, but I didn't know why. Thanks for so bluntly pointing it out.

Sorry about that, I know I lack tact. Please feel free to point that out. It helps me to know when I should apologize for my bluntness.

1. Philosophy Forum
2. » Logic
3. » If ZFC is incomplete it can not prove anything