Do Gödel's incompleteness theorems imply Platonism is true of our world?

[YHWH_Christ]

Gödel’s incompleteness theorems state:

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F ."

and

"Assume F is a consistent formalized system which contains elementary arithmetic. Then {\displaystyle F\not \vdash {\text{Cons}}(F)}

Gödel demonstrated these theorems through his genius of Gödel numbering which can be done in an infinite amount of ways (Gödel uses unique prime factorization to demonstrate this) and essentially makes them become self referential, i.e. if a formal system is consistent (all statements in the system are provable within the system) then there must exist a statement from the generated strings which is not provable, thus its provability demonstrates it unprovability making it incomplete. This means there are statements, such as 0+1=1 is true, that are true but which cannot be proven except within themselves. Would this then imply at least Mathematical Platonism?

 

[LeonardDeNoblac]

I’m terrible at maths, but I’ve heard about this theorem before, and from what I’ve understood it demonstrates that there can be no complete and consistent mathematical non-banal formal system. Since physics is largely about describing reality with mathematical models, I think that the theorem can demonstrate that, since every mathematical non-banal system is either incomplete or inconsistent, reality can’t be accurately described in merely physical terms.
But maybe I’m wrong, anyone can feel free to correct me, as I’ve said I’m terribile at maths.

 

[YHWH_Christ]

For me this implies that in order to describe physical systems, like what Physics attempts to do, non physical operations must be used. Is the physical world then emergent from a non physical world that is based on logic? The fact that we know the laws of logic to be true, yet not completely provable, seems to also hammer that idea home. Kurt Gödel himself was a Platonist and I can see why.

 

[LeonardDeNoblac]

The laws of logic are true because their negation would imply absurdities. It’s not the same way other principles are demonstrated, but isn’t it still a proof in some way?

 

[YHWH_Christ]

Well you can have negations, to quantify something though is give it a quantifier. The negation of one thing could imply another

 

[tafan2]

The laws of logic are True but not provable us not what Godel’s theorem states.

Rather, in any mathmatical system at least as Least as sophisticated as number theory, there are true statements that cannot be proven within that theory.

 

[YHWH_Christ]

That’s basically what I said.

 

[Dovekin]

The laws of logic are True but not provable us not what Godel’s theorem states.

Rather, in any mathmatical system at least as Least as sophisticated as number theory, there are true statements that cannot be proven within that theory.

Are the laws of logic a mathematical system at least as sophisticated as number theory? THis seems to be one of the questions.

Another question: If we decide to describe the physical world using a mathematical system, what are the implications of Gödel’s theories? Is the physical world dependent on the mathematical description? Or put another way, is the physical world bound by the same principles that bind our mathematical models?

 

[tafan2]

Are the laws of logic a mathematical system at least as sophisticated as number theory? THis seems to be one of the questions.

They would be part of any such system. But it is the mathmatical system, nit the logic system which the theory applies.

 

[tafan2]

If we decide to describe the physical world using a mathematical system, what are the implications of Gödel’s theories?

Relative to laws of physics and such describing our world? Not much. Oh, I suppose you could come up with some formula that describes something, and it wouldn’t be provable due to the incompleteness of the mathmatical system being used, but that us both unlikely and if it happened, just branch out of that system.

Godel figured out how to say " number theory is consistent" ( ie there are no statements which can be proven true and false) with number theory itself. That was quite an accomplishment, but then he showed that number theory could not prove that statement to be true. Rather esoteric, and only if interrst to logicans until computers came along.