The Instigator
shakuntala
Pro (for)
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The Contender
Enji
Con (against)
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Godels theorem ends in meaninglessness

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Voting Style: Open Point System: 7 Point
Started: 7/3/2013 Category: Science
Updated: 3 years ago Status: Post Voting Period
Viewed: 1,237 times Debate No: 35247
Debate Rounds (3)
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shakuntala

Pro

This author points out
http://www.scribd.com...

GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
thus his incompleteness theorem ends in meaninglessness

Now the syntactic version of Godels first completeness theorem reads
Proposition VI: To every `9;-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r).

But when this is put into plain words we get

http://en.wikipedia.org...

"Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250)

In other words there are true mathematical statements which cant be proven
But the fact is Godel cant tell us what makes a mathematical statement true thus his theorem is meaningless

Now
Peter Smith the Cambridge expert on Godel admitts
http://groups.google.com...

Quote:
Godel didn't rely on the notion
of truth

but truth is central to Godel's theorem
".... any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true... "(Kleene 1967, p. 250)

you see Godel refers to true statements
but Godel didn't rely on the notion as the Cambridge expert Peter Smith says

now because Godel didn't rely on the notion of truth he cant tell us what true statements are
thus his theorem is meaningless

If Godels theorem said there were gibbly statements that cant be proven
But if Godel cant tell us what a gibbly statement was then we would say his theorem was meaningless

if you disagree
then just tell us what Godel said makes a mathematical statement true
Enji

Con

Godel's incompleteness theorems are, perhaps, some of the most interesting proofs regarding the philosophy of mathematics (I think so, anyway). Unfortunately, my opponent clearly misunderstands what is proven. For clarity, I will be using a simpler, easier to read version of the proof.

What the proof says:

Given a computably generated set of axioms, let PROVABLE be the set of numbers which encode sentences which are provable from the given axioms.
Thus for any sentence s,
(1) < s > is in PROVABLE iff s is provable.
Since the set of axioms is computably generable, so is the set of proofs which use these axioms and so is the set of provable theorems and hence so is PROVABLE, the set of encodings of provable theorems.
Since computable implies definable in adequate theories, PROVABLE is definable.
Let s be the sentence "This sentence is unprovable".
By Tarski, s exists since it is the solution of:
(2) s iff < s > is not in PROVABLE.
Thus
(3) s iff < s > is not in PROVABLE iff s is not provable.
Now (law of excluded middle) s is either true or false.

If s is false, then by (3), s is provable.
This is impossible since provable sentences are true.
Thus s is true.
Thus by (3), s is not provable.
Hence s is true but unprovable.
[http://www.math.hawaii.edu...]

________________________________________________________

PROVABLE is the set of all provable statements, and if a statement can be proven using the axioms of the system, then it is contained in the set PROVABLE. In mathematics, the axioms of the system are accepted to be true and if a mathematical statement can be proven using the axioms of the system then that mathematical statement is accepted to be true; hence PROVABLE is a subset of TRUTH (the set of all statements accepted to be true). The statement s, "s is not provable", is true if and only if it is not in PROVABLE and s is not in PROVABLE if and only if s is not provable (3).

s is either true or false (law of excluded middle); if s is false, then s is in PROVABLE (3) thus s is provable - however since provable statements are accepted to be true, this is impossible - the system would be inconsistent. Thus s is true and therefor s is not provable, therefor PROVABLE is not the the equivalent of TRUTH; the system is incomplete.

Why the meaning of truth is irrelevant:

As my opponent's source Peter Smith says, Godel doesn't rely on the notion of truth, but rather the notion of proof. If a system cannot prove a statement nor its denial, then it is incomplete - regardless of whether the axioms of the system are true. Godel's proof shows that any consistent mathematical system capable of basic arithmetic cannot prove all statements nor their denials, therefor any such system is incomplete. Hence why it's called "Godel's incompleteness theorem".
Debate Round No. 1
shakuntala

Pro

Con Did not answer my question ie what Godel says makes a mathematical statement true

you say
"Why the meaning of truth is irrelevant:

As my opponent's source Peter Smith says, Godel doesn't rely on the notion of truth, but rather the notion of proof. If a system cannot prove a statement nor its denial, then it is incomplete - regardless of whether the axioms of the system are true. Godel's proof shows that any consistent mathematical system capable of basic arithmetic cannot prove all statements nor their denials, therefor any such system is incomplete. Hence why it's called "Godel's incompleteness theorem"."

Godels proof is about there being true statements which cant be proven
".... any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true... "(Kleene 1967, p. 250)

if Godel cant tell us what makes a mathematical proof true then his theorem is meaningless

do you accept this statement about what Godels theorem proved

"Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250)

if yes
then tell us what Godel says makes a mathematical statement true
if you cant then Godels theorem is meaningless
Enji

Con

A statement is either true or false. If TRUTH (the set of all true statements) is a subset of PROVABLE (the set of all statements which can be proven by the axioms of a mathematical system) and PROVABLE contains statements which are not contained in TRUTH, then PROVABLE contains statements which are false - thus the system is inconsistent.

If PROVABLE is a subset of TRUTH and TRUTH contains statements which are not contained in PROVABLE, then PROVABLE does not contain all true statements - thus the system is incomplete.

Godel's incompleteness theorem proves that any mathematical system capable of basic arithmetic is necessarily either inconsistent or incomplete. Since either a statement or its denial are true, Godel's incompleteness theorem shows that for any consistent mathematical system there exists true statements which cannot be proven to be true.

We can flip the meaning of truth and this still holds - as long as a mathematical system is consistent and capable of basic arithmetic, the system is incomplete regardless of whether the axioms of the system are true.
Debate Round No. 2
shakuntala

Pro

Con answers
"A statement is either true or false. If TRUTH (the set of all true statements) is a subset of PROVABLE (the set of all statements which can be proven by the axioms of a mathematical system) and PROVABLE contains statements which are not contained in TRUTH, then PROVABLE contains statements which are false - thus the system is inconsistent.

"If PROVABLE is a subset of TRUTH and TRUTH contains statements which are not contained in PROVABLE, then PROVABLE does not contain all true statements - thus the system is incomplete.

Godel's incompleteness theorem proves that any mathematical system capable of basic arithmetic is necessarily either inconsistent or incomplete. Since either a statement or its denial are true, Godel's incompleteness theorem shows that for any consistent mathematical system there exists true statements which cannot be proven to be true.

We can flip the meaning of truth and this still holds - as long as a mathematical system is consistent and capable of basic arithmetic, the system is incomplete regardless of whether the axioms of the system are true."

I have presented case- Godel's theorem is meaningless as he cant tell us what makes a mathematics statement true
I have backed up my case by citing a Cambridge expert ie Godel did not use a notion of truth

Con says
"then PROVABLE does not contain all true statements"

but she does not tell us what Godel says make a true mathematics statement

thus her statement
"then PROVABLE does not contain all true statements"
is meaningless as we dont know what "true statements" are
and Godel's theorem is meaningless as Godel cant tell us what make a true mathematics statement
Enji

Con

Since either a statement or its negation are true (law of excluded middle) and Godel's proof shows that neither the statement nor its negation can be proven to be true while remaining consistent with the axioms, what makes a mathematical statement true is irrelevant to Godel's proof. Hence Peter Smith's statement that Godel's incompleteness theorem does not concern itself with truth - truth is irrelevant to the system's incompleteness.

Godel's incompleteness theorem shows that no consistent mathematical system capable of elementary arithmetic can prove either all statements or their negations to be true, thus all such systems are incomplete. Thus, Godel's first incompleteness theorem is meaningful.
Debate Round No. 3
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