Total Posts:5|Showing Posts:1-5
Jump to topic:

Godels theorem ends in meaninglessness

shakuntala
Posts: 32
Add as Friend
Challenge to a Debate
Send a Message
7/3/2013 7:59:21 AM
Posted: 3 years ago
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
drafterman
Posts: 18,870
Add as Friend
Challenge to a Debate
Send a Message
7/3/2013 9:04:25 AM
Posted: 3 years ago
First, you start with your axioms. A set of propositions and statements that we accept, as a given, as "true."

Second, we have our rules of inference. A set of rules that we accept, as a given, as valid such that: a truth + a rule of inference = a truth.

Third, and lastly, we have all truths as a sum of our axioms combined with all of the valid combinations of our rules of inference. We will call the set of such truths "T."

Now, take the following statement:

"This statement is not a member of 'T'"

Either that statement *is* in T, which means T contains a falsehood and is inconsistent, or that statement *isn't* in T, which mean T doesn't contain a truth and is incomplete.
chui
Posts: 507
Add as Friend
Challenge to a Debate
Send a Message
7/3/2013 9:11:02 AM
Posted: 3 years ago
Maths is based on a set of axioms. These are rules that are held to be true but which cannot be proven. The axioms are usually expressed in terms of set theory because natural numbers can be expressed as sets without need to use analogies or comparisons to objects.
Using the full set of axioms it is possible to prove many statements true. eg a x b = b x a for any value of a or b. Everything in maths that has been shown to be true is still true.
What Godel proved is that there are some statements that are true that have no proof. For example:- "this statement has no proof that it is true". It can only be disproven by proving it, so disproving is logically impossible so it must be true so has no proof.
So far all of the statements that are found to be non-provable but true are irrelevant, such as the example above. So maths is still meaningful.
tBoonePickens
Posts: 3,266
Add as Friend
Challenge to a Debate
Send a Message
7/8/2013 8:37:56 AM
Posted: 3 years ago
At 7/3/2013 9:11:02 AM, chui wrote:
Maths is based on a set of axioms. These are rules that are held to be true but which cannot be proven. The axioms are usually expressed in terms of set theory because natural numbers can be expressed as sets without need to use analogies or comparisons to objects.
Using the full set of axioms it is possible to prove many statements true. eg a x b = b x a for any value of a or b. Everything in maths that has been shown to be true is still true.
What Godel proved is that there are some statements that are true that have no proof. For example:- "this statement has no proof that it is true". It can only be disproven by proving it, so disproving is logically impossible so it must be true so has no proof.
So far all of the statements that are found to be non-provable but true are irrelevant, such as the example above. So maths is still meaningful.
The conclusion drawn from Godel isn't that math isn't meaningful.
WOS
: At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote:
: Without nothing existing, you couldn't have something.