# if set theory ie ZFC is incomplete it cannot prove anything

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3. » if set theory ie ZFC is incomplete it cannot prove anything

Mon 12 May, 2008 09:03 pm
the australian philosopher colin leslie dean points out that the systems ZFC, PA, Q due to there, incompleteness cant 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

this follows from Colin leslie dean who argues that Mathematics is
systems of epistemological holisim
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

Nocturne

Tue 3 Jun, 2008 12:36 am
@pam69ur,
pam69er,

[indent]1. ZFC is used to prove things in mathematics
2. ZFC can only prove things if all its statement can be proven to be true
3. ZFC has statements which cannot be proven true or false

Therefore,

4. ZFC cannot prove anything[/indent]

The second premise is false. It is only necessary that ZFC is consistent for it to be useful in the construction of proofs, not that every and each of 'its statements can be proven to be true' (whatever that means exactly). The validity of an inference is in no way dependent on our ability to prove that the inference is valid, and its consistency (or lackthereof) does not change in response our discoveries and investigations, but is constant and independent of what we think about it. In other words, if we suppose that tomorrow your second premise is satisfied so that ZFC can be used in proof construction, then it must also be good for the construction of proofs right here and now (i.e. if every inference is valid after the second premise has been satisified, then so every inference beforehand must also have been valid). Therefore, it is not necessary that your second premise is fulfilled before ZFC can be used in proof construction, since ZFCs consistency is a property independent of what we can or cannot prove about ZFC or anything else.

VideCorSpoon

Tue 3 Jun, 2008 07:26 am
@Nocturne,
Nocturne,

pam69ur has been going 'round the internet under different alias trying to convey ideas he (or she for that matter) does not understand. Refer to the post "mathematicians are in big trouble for 2 reasons" near the end. Justin and other moderators think it's he who must not be named, the Australian philosopher coli...Ack!!! I can't breath!!!! My lack of faith has indeed been disturbing!!!!

Also, I don't know if the moderators kicked him off or not, so I don't know if you'll get a response.

pam69ur

Tue 10 Jun, 2008 08:13 am
@VideCorSpoon,
Quote:
Therefore, it is not necessary that your second premise is fulfilled before ZFC can be used in proof construction, since ZFCs consistency is a property independent of what we can or cannot prove about ZFC or anything else

complete rubbish

take logic
logic is a system of laws
1)law of idenity
3)law of excluded middle

you use these laws in your maths proofs
but unless these laws can be proven to be true
then you cannot prove anything with them
take the law of identity
that law impiles that there is an essence

Quote:
[1] In this regard the law of identity is the ultimate foundation upon which logic rests, without an 'identity' (for the symbols of logic) logic is overthrown and collapses.......The law of non-contradiction to quote Aristotle states " the same attribute (characteristic essence) cannot at the same time belong and not belong to the same subject and in the same respect."[3] In terms of propositional calculus ' it is not the case both p and not p'. In this regard we see that if there is no essence to characterise a subject in distinction from other subjects there can be no law of non-contradiction and thus no logic at all. In other words if there is nothing to distinguish a 'horse' from a 'non-horse', either ontological or nominal, in the proposition P 'there is a horse' then we can not apply the law of non-contradiction because we have no distinguishable subject for the subject of the proposition.

[1] W. R. B, Gibson, 1908, p,95.

[2] C, Dean, op. cit. p. XXV-XXXV.

[3] A, Flew, 1979, p.75.

now untill this essence is proven the law of identity thus law of non-contradiction cannot be used to prove anything all i can do with these laws is offer conjectures

similary
if i use 1+1=2 in a proof
untill i can prove 1+1=2 then i cannot prove anything with it all i can do is give a conjecture with it

Conjecture - Wikipedia, the free encyclopedia

Quote:
In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic

Professer Frost

Tue 10 Jun, 2008 08:50 am
@pam69ur,
Who the heck is Colin Dean and what has he done worthy of note other than (apparently) pay people to go online promote his "philosophy" without let up?
- The Prof.

VideCorSpoon

Tue 10 Jun, 2008 09:33 am
@Professer Frost,
Prof. Frost,

pam69ur has been going 'round the internet under different alias trying to convey ideas he (or she for that matter) does not understand. Refer to the post "mathematicians are in big trouble for 2 reasons" near the end. Justin and other moderators think it's he who must not be named, the Australian philosopher coli...Ack!!! I can't breath!!!! My lack of faith has again indeed been disturbing!!!!

But I guess he's still around if he replied. Case in point, go look at some of pam69ur's other posts in the logic section for a bigger picture of what's going on.

Professer Frost

Tue 10 Jun, 2008 10:31 am
@VideCorSpoon,
I can't understand mathematical logic but after skimming a few threads it would appear that neither can this worshiper of ye omniscient philosopher Colin ****** Dean.
- The Prof.

Nocturne

Tue 10 Jun, 2008 03:09 pm
@pam69ur,
pam69ur,

[indent]1. If the laws of identity, noncontradiction and excluded middle cannot be proven then they cannot be used to prove anything else.[/indent]

The laws of identity, noncontradiction and excluded middle are generally assumed by the deductive apparatus, and so to prove them is simply to state explicitly what has been assumed implicitly. It would be a proof in the mathematical sense, but not a 'proof' in the sense which you seem to want, because it would evidently beg the question i.e. if you define 'truth' and 'falsity' so that a proposition can only be true or false, but not both or anything else, then the law of excluded middle clearly follows... but then what have you really proven?

[indent]2. The law of identity implies that there is an essence.[/indent]

False. The law of identity implies that for every x, whatever x might be, x is equal to x. To infer from the law of identity that x has an essence is invalid. The inference would only be valid if you are using the words 'identity' and 'essence' synonymously, in which case you are simply repeating yourself and saying nothing of interest or importance.

In any case, my original point was the validity of an inference does not depend on our ability to prove that it is valid, just as the truth of a proposition does not depend on our ability to prove that it is true i.e. if a proposition true, then it will be true before we have proven that it is true, after we have proven that it is true, and even if we cannot prove that it is true. For example, if I use '1 + 1 = 2' in a mathematical proof, then it is expected that I can or will also develop a proof for '1 + 1 = 2', but the validity of my argument does not depend on whether I have proven '1 + 1 = 2', but only on whether '1 + 1 = 2' is a valid inference. In other words, the validity of an argument depends upon whether or not the rules of inference used are truth-preserving, and not on whether those rules have been proven to be truth-preserving.

pam69ur

Tue 10 Jun, 2008 06:00 pm
@Nocturne,
Quote:

In any case, my original point was the validity of an inference does not depend on our ability to prove that it is valid, just as the truth of a proposition does not depend on our ability to prove that it is true i.e. if a proposition true, then it will be true before we have proven that it is true, after we have proven that it is true, and even if we cannot prove that it is true. For example, if I use '1 + 1 = 2' in a mathematical proof, then it is expected that I can or will also develop a proof for '1 + 1 = 2', but the validity of my argument does not depend on whether I have proven '1 + 1 = 2', but only on whether '1 + 1 = 2' is a valid inference. In other words, the validity of an argument depends upon whether or not the rules of inference used are truth-preserving, and not on whether those rules have been proven to be truth-preserving.

complete rubbish
wiles proved fermats last theorem with the Taniyama-Shimura conjecture,
untill this conjecture was proven he could not give a proof of fermats last theorem
all he could do untill the conjecture was proven was use it to give another conjecture

same with 1+1=2 untill it is proven all you can do with it is giive conjectures

same with ZFC
with unprovable statements
ZFC cant prove anything but only give conjectures

Quote:
In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture. (His article was published in 1990.) He demonstrated that, just as Frey had anticipated, a special case of the Taniyama-Shimura conjecture (still unproven at the time), together with the now proven epsilon conjecture, implies Fermat's Last Theorem. Thus, if the Taniyama-Shimura conjecture holds for a class of elliptic curves called semistable elliptic curves, then Fermat's Last Theorem would be true.
After learning about Ribet's work, Andrew Wiles set out to prove that every semistable elliptic curve is modular. He did so in almost complete secrecy, working for a full seven years with minimal outside help. Over the course of three lectures delivered at Isaac Newton Institute for Mathematical Sciences on June 21, 22, and 23 of 1993, Wiles announced his proof of the Taniyama-Shimura conjecture, and hence of the Fermat's Last Theorem. Wiles drew upon a wide variety of methods in the proof, some of them having been developed especially for this occasion.

Nocturne

Tue 10 Jun, 2008 06:41 pm
@pam69ur,
Pam69ur,

I agree. Moreover, I think that all knowledge is conjectural, and that there is never any justification for any argument, proof, or rule of inference. However, that has nothing to do with whether an argument, proof, or rule of inference is valid, and I do think that some are. The term 'proof' in mathematics is a hangover from a time when mathemtical arguments were expected or intended to actually prove something, and that they might not would be considered a problem. For example, let us agree that only conjectures can be had from ZFC. That a statement or argument is conjectural does not imply that it is false or invalid, and it is not a criticism of such a statement of argument that it might be false or invalid. If you actually have a criticism of ZFC, rather than the trivial fact that it could possibly be mistaken, please provide it.

pam69ur

Tue 10 Jun, 2008 06:57 pm
@Nocturne,
Quote:
That a statement or argument is conjectural does not imply that it is false or invalid, and it is not a criticism of such a statement of argument that it might be false or invalid.

complete besides the point
wiles if he offered a proof of the fermats theorem with an unproven Taniyama-Shimura conjecture it would not have been accepted as a proof at all

same applies to ZFC if there are statements which canot be proven -like the once
Taniyama-Shimura conjecture- then ZFC cant be used to prove anything but only give conjectures

Quote:
If you actually have a criticism of ZFC, rather than the trivial fact that it could possibly be mistaken, please provide it.

i have given it
same applies to ZFC if there are statements which canot be proven -like the once Taniyama-Shimura conjecture- then ZFC cant be used to prove anything but only give conjectures

if it applies to wiles proof
it then applies to ZFC
ie with out proof of all the statements -ie
Taniyama-Shimura conjecture- then ZFC cant proove anything

Professer Frost

Tue 10 Jun, 2008 08:54 pm
@pam69ur,
Pam69ur,
Look, if you want to start a thread on the overall philosophy of Colin L. Dean go ahead, but I get the distinct impression that everyone on the forum is sick of your endless harping on this particular topic.
- The Prof.

Aedes

Tue 10 Jun, 2008 09:06 pm
@pam69ur,
pam69ur wrote:
same with 1+1=2 untill it is proven all you can do with it is giive conjectures
What's to prove? 1+1=2 is a completely closed tautology.

Nocturne

Wed 11 Jun, 2008 09:57 am
@pam69ur,
pam69ur,

The hypotheses of science will be forever unproven. That is, the past success of a scientific hypothesis does not imply its future success, and so by convention scientists hold every scientific hypothesis open to refutation. However, the elementary fact that scientific hypothesis are unproven is not generally considered a cause for alarm, and neither is it thought a damning criticism. In other words, the unprovable status of scientific hypotheses does not imply that they are false, and since scientific investigation is concerned with discovering true theories, and not provably true theories, the unprovability of scientific theories is not something to be too worried about.

In a like manner, there are statements about some formal languages which cannot be proven to be true by that language--else the language must be inconsistent. Therefore, if a language is consistent then it is impossible to construct a proof for every statement of that language, and so mathematicians, like scientists, by convention hold their hypotheses open to refutation. However, that it is impossible to prove, once and for all, that such a language is consistent is not a cause for alarm, and neither is it a damning criticism. In other words, the unprovable status of some mathematical languages does not imply that they are inconsistent, and since it is their consistency which we are concerned with, that they might be unprovable is not something to be too worried about.

pam69ur

Wed 11 Jun, 2008 09:16 pm
@Nocturne,
Quote:
The hypotheses of science will be forever unproven. That is, the past success of a scientific hypothesis does not imply its future success, and so by convention scientists hold every scientific hypothesis open to refutation. However, the elementary fact that scientific hypothesis are unproven is not generally considered a cause for alarm, and neither is it thought a damning criticism. In other words, the unprovable status of scientific hypotheses does not imply that they are false, and since scientific investigation is concerned with discovering true theories, and not provably true theories, the unprovability of scientific theories is not something to be too worried about.

In a like manner, there are statements about some formal languages which cannot be proven to be true by that language--else the language must be inconsistent. Therefore, if a language is consistent then it is impossible to construct a proof for every statement of that language, and so mathematicians, like scientists, by convention hold their hypotheses open to refutation. However, that it is impossible to prove, once and for all, that such a language is consistent is not a cause for alarm, and neither is it a damning criticism. In other words, the unprovable status of some mathematical languages does not imply that they are inconsistent, and since it is their consistency which we are concerned with, that they might be unprovable is not something to be too worried about.

all irrelevant

as colin ****** dean has noted

ZFC cannot prove anything as it has statements which cant be proven

in ZFC if there are statements which canot be proven -like the once Taniyama-Shimura conjecture- then ZFC cant be used to prove anything but only give conjectures

if it applies to wiles proof
it then applies to ZFC
ie with out proof of all the statements -ie
Taniyama-Shimura conjecture- then ZFC cant proove anything

if wiles gave a proof using an unproven
Taniyama-Shimura conjecture-mathematicians would have said he did not give a proof as one of his statements was itself unproven

so
mathematicans must then say ZFC cant give any proof as some of its statements are unproven
VERY SIMPLE
what applies to wiles must then apply to ZFC

Nocturne

Fri 13 Jun, 2008 01:15 pm
@pam69ur,
pam69ur,

I agree. However, this is only a problem if you are attempting to achieve a secure proof, something which I am not particularly concerned with. In short, that a derivation may not be a proof in the sense which you mean does not mean that the same derivation is invalid, and so until someone produces an critical argument which purports to show that a language is inconsistent, I am not going to hold the mere possibility that it is inconsistent against it. In other words, the problem which seems to concern you is not inherent to mathematics, but with what you want to do with mathematics, and since I do not share the same goals I do not share the problem which afflicts you. Therefore, the result of the Godelian theorem doesn't concern me in the same way that it concerns you, since what you seek to do by mathematical argument is not something which I am interested in achieving.

If you want to argue that ZFC, or any other formal language is inconsistent then I am all ears. If, however, you simply want to rant on about how "proofs" in these languages do not provide the security which you would like then I am not interested--that is your problem, not mine.

VideCorSpoon

Fri 13 Jun, 2008 02:36 pm
@pam69ur,
pam69ur wrote:
all irrelevant

as colin ****** dean has noted

ZFC cannot prove anything as it has statements which cant be proven

in ZFC if there are statements which canot be proven -like the once Taniyama-Shimura conjecture- then ZFC cant be used to prove anything but only give conjectures

if it applies to wiles proof
it then applies to ZFC
ie with out proof of all the statements -ie
Taniyama-Shimura conjecture- then ZFC cant proove anything

if wiles gave a proof using an unproven
Taniyama-Shimura conjecture-mathematicians would have said he did not give a proof as one of his statements was itself unproven

so
mathematicans must then say ZFC cant give any proof as some of its statements are unproven
VERY SIMPLE
what applies to wiles must then apply to ZFC

Here we go again...

Almagor

Sat 19 Mar, 2011 10:30 pm
Pam69ER is correct that if a theory has not been proved correct it may be correct before it has been proved correct. Pam 69Er omits that this is only that it "may" be correct, it also may be incorrect. We don't know until it is proved to be one or the other. To use a theory in the real world that has not been proved correct as Pam69Er suggests when it may just as easily be incorrect may get a person in a lot of trouble!. If a person drives their truck across a bridge that has not been proved to be able to support the truck but "might" or "maybe" be strong enough to support the weight of the truck, both the truck and the driver may be lost as the truck plummets to the bottom of the 1000 foot deep gorge because "maybe" is not good enough when life and death are the issue.

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