Philosophy Now Forum

The Penrose-Lucas argument:

https://en.wikipedia.org/wiki/Penrose%E ... s_argument

Due to human ability to see the truth of formal system's Gödel sentences, it is argued that the human mind cannot be computed on a Turing machine that works on Peano arithmetic because the latter cannot see the truth value of its Gödel sentence, while human minds can.

How do humans know that a mathematical sentence is true? There is only one way: by proving it. Otherwise, it will be deemed a hypothesis and not a (true) theorem. But then again, we are still able to correctly detect some Gödel sentences, i.e. sentences that are true but not provable, but that requires a rather special situation, such as for example, in the case of the Goodstein's theorem.

The language in which Goodstein's theorem is phrased, is Peano Arithmetic theory (PA). However, the language in which its proof is phrased, is Zermelo-Fränckel set theory (ZF). Its proof uses infinite ordinals, which are defined in ZF but not in PA. Hence, Goodstein's theorem belongs to PA but its proof does not belong to PA. Its proof belongs to ZF. That is why we know that Goodstein's theorem is true in ZF and therefore also in PA. Hence, from the standpoint of PA, Goodstein's theorem is indeed true but not provable, i.e. a Gödel sentence.

A Turing machine could also use ZF to prove an otherwise unprovable theorem in PA. Therefore, it is not something that only human minds can do. What if there is no alternative theory available to prove the Gödel sentence from? In that case, both humans and the machine will not be able to know that the Gödel sentence is true. They will both consider it to be just a hypothesis.

Conclusion. The ability to see the truth of Gödel sentences is not different between human minds and Turing machines. In the general case, they will both fail to do it. The human mind may still be superior to Turing machines but not for its ability to see the truth of Gödelian sentences.

"Goodstein's theorem is true in ZF and therefore also in PA"

Whoa, back up the wagon, Chester! Does Goodstein's theorem prove that a proposition which is true in ZF must also be true in PA? We know that ZF underwrites PA in many respects, but to assert that a ZF proposition is automatically true in PA is drawing a long bow. In fact, it's demonstrably wrong. PA tells us that 0/2=0; ZF tells us it's undefined. Can you cite arguments to support your position? I admit I have no idea exactly what Goodstein asserts; I only want to clarify the above assertion.

godelian wrote: ↑Sat May 04, 2024 3:43 pm The Penrose-Lucas argument:

Not another argument! Why can't people learn to put their differences aside?

alan1000 wrote: ↑Sat Jul 06, 2024 7:32 pm "Goodstein's theorem is true in ZF and therefore also in PA"

Whoa, back up the wagon, Chester! Does Goodstein's theorem prove that a proposition which is true in ZF must also be true in PA? We know that ZF underwrites PA in many respects, but to assert that a ZF proposition is automatically true in PA is drawing a long bow. In fact, it's demonstrably wrong. PA tells us that 0/2=0; ZF tells us it's undefined. Can you cite arguments to support your position? I admit I have no idea exactly what Goodstein asserts; I only want to clarify the above assertion.

Claim: If the proposition can be expressed in PA and it is provable in ZF, it will be true in PA's models.

This should be true because PA is bi-interpretable with ZF-inf.

It amounts to stating that if the proposition can be expressed in ZF-inf and it is provable in ZF, that it will be true in ZF-inf's models.

I think that we may be able to use the compactness theorem for this purpose.

There is no human mind.

And for those who think or believe there is, then define the word 'mind', inform the readers how you, human beings, have/own/possess 'this thing' that 'your definition' refers to, and then explain how you 'know' what you claim here is 'irrefutably True', when 'it' cannot be 'observed nor experienced with any of the five senses of the human body.

To the OP. No, common sense suffices.