Does Godel's theorem imply mathematics is more than logic?

[nonequilibrium]

When reading about Gödel's incompleteness theorem(s) a few years back, I vaguely remember reading the statement that one of their implications was that mathematics can not really be seen as "merely" logic. I think the reasoning was something like: since we have sentences "this cannot be proven within our logical system" and we know they are true, and knowing they are true is conceptually a proof yet on a more meta-level (since you can't strictly prove it in the logical system), the situation is in shorthand: we have statements that can be proven in mathematics, but that can't be proven in logic.

I know that sounds wishy-washy, and don't misunderstand me: I'm not trying to present that as an argument and starting a debate. Rather I'm just trying to remember what I had read and I was wondering whether this sounds familiar to someone? I was trying to find a discussion of this online but couldn't find anything. (It might be that Penrose makes such a claim(?))

 

[bahamagreen]

Your example implies to me that logic would be "more" than mathematics... in the sense of more fundamental in the hierarchy.
This is a little different than saying "math is more than logic" in the sense that math might include things outside or beyond logic, or elements that can't be derivered or traced back to logic.

 

[jgens]

I think the reasoning was something like: since we have sentences "this cannot be proven within our logical system" and we know they are true, and knowing they are true is conceptually a proof yet on a more meta-level (since you can't strictly prove it in the logical system), the situation is in shorthand: we have statements that can be proven in mathematics, but that can't be proven in logic.

This is one of the most annoyingly persistent misconceptions about the incompleteness theorems. The problem stems the different notions of completeness in mathematical logic:

1. A deductive system is called complete when every semantically provable sentence is syntactically provable (the converse is called soundness and any decent deductive system will be sound). In heuristic terms, the semantically provable sentences are the "true" sentences, while the syntactically provable sentences are those sentences having a formal proof in our deductive system. This notion of completeness is relevant for statements like the Gödel completeness theorems which assert that first-order logic (for example) is both sound and complete with this meaning.
2. A formal theory is called complete when it consists of a maximal set of consistent sentences. This means that given any sentence in our language our theory contains either a proof of this sentence or a proof of its negation. This notion of completeness is relevant for statements like the Gödel incompleteness theorems which assert that any sufficiently strong theory is either incomplete or inconsistent. This has almost nothing to do with "true" sentences being unprovable. Rather it means that (assuming consistency) there are statements whose truth value we cannot determine. In ZF set theory the axiom of choice is an example of such a statement. There are models of ZF where the axiom choice holds and yet other models of ZF where it fails.

In short whoever is telling you that the incompleteness theorems concern "true" theorems without formal proofs is getting their notions of completeness mixed up.

 

[jgens]

It was pointed out the above needs modification. See the content in the following post for the correct information: https://www.physicsforums.com/showthread.php?t=733432#post4634065. In short the issue comes down to what is meant by "true statements" concerning arithmetic. What people have been meaning is "true statements in a very specific model of the natural numbers" instead. Sorry for the misinformation

 

[blue_raver22]

I can provide some insight into this topic. First, it is important to understand what Gödel's incompleteness theorems actually state. These theorems say that any consistent formal system of mathematics cannot prove all true statements within that system. In other words, there will always be statements that are true but cannot be proven within that system.

So, does this mean that mathematics is more than logic? The answer is both yes and no. On one hand, Gödel's theorems show that mathematics is not reducible to a purely logical system. It requires additional axioms and principles to prove certain statements. On the other hand, mathematics is still based on logical reasoning and follows logical rules.

One way to think about this is that mathematics is a language and logic is its grammar. Just like how we use grammar rules to construct sentences in a language, we use logical rules to construct mathematical statements and proofs. However, there are certain ideas and concepts in mathematics that cannot be fully captured by logic alone. This is where additional axioms and principles come in to fill the gaps.

In summary, Gödel's theorems do imply that mathematics is more than logic in the sense that it requires more than just logical rules to prove all true statements. However, logic still plays a fundamental role in mathematics and cannot be separated from it.