What’s the consequences of Godel incompleteness theorem?
In: Mathematics
The other explanation here is pretty good, although I would nitpick a little: the Incompleteness theorems do **not** say that we will be unable to prove some things that “are true,” since a statement by definition cannot be true in an axiomatic system if it cannot be proven from that axiomatic system.
A better way of phrasing this is to say that the Incompleteness theorems imply that, in any consistent and sufficiently rich axiomatic system (i.e. any set of axioms that do not allow you to prove contradictory statements and that are complex enough to model arithmetic), there are some statements that are undecidable, meaning that, while we can state the statement in the language of the axiomatic system, we can prove neither the statement nor its negation from the axioms.
An example in the axiomatic system ZFC (which is the standard axiomatic system for set theory and lies at the foundation of most modern mathematics) is the Continuum Hypothesis, which is the statement that there exists some set whose cardinality is strictly between the cardinality of the integers and the cardinality of the reals. We cannot prove CH to be true or false from ZFC (and we can prove that this is the case!). Furthermore, if we were to add on to our current set of axioms (ZFC) either the assumption that the CH is true or the assumption that CH is false, we would still get a consistent system; that is, our axiomatic system “doesn’t say anything” about CH and doesn’t care whether or not it’s true.
For a list of some other statements that are provably undecidable in ZFC, see https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC.
One statement that might catch your eye on that list is the consistency of ZFC—indeed, we can prove that we cannot prove, in ZFC, whether ZFC is consistent or not! (This is the second Incompleteness theorem of Gödel, where he showed that if ZFC can prove that ZFC is consistent, then that implies that ZFC is not consistent.) But this is actually not a huge deal. For more on this, see the answers on this page: https://math.stackexchange.com/questions/1746563/why-is-establishing-absolute-consistency-of-zfc-impossible. In short, there is a real difference between truly formal proofs (which use rules of inference and a set of axioms to prove statements in a formal system) and the sort of proofs that most mathematicians actually do on a daily basis, which are far more high-level than truly formal proofs. So the fact that no formal proof exists of certain statements really doesn’t end up being a huge deal. Unfortunately, at this point things become very, very technical very fast, and it’s basically impossible to ELI5. (Just look up “cumulative hierarchy” or “reflection principle” to see how complex things get.)
There are also some possibly very simple examples. Suppose I define an axiomatic system that defines the notion of a car and the notion of color, and the notion that a car may be a certain color. Then let X be a car and consider the statement “X is red.” This statement is undecidable from my axiomatic system, since X could be red or could be not red, and either way no contradiction emerges. That is, I could create an axiomatic system where X is red, or one where X is not red, and both would be consistent.
Note that the Incompleteness theorems apply not just to ZFC, but to any axiomatic system meeting certain requirements, which are too technical to ELI5 without oversimplifying.
And one last note: The Incompleteness theorems absolutely do not mean that “math is beyond human comprehension” or that there are “some mysteries in math that we will never find out” or any such thing. They have very precise implications in terms of the properties of axiomatic systems, and any grandiose statements like the above are just gross misinterpretations of the precise results. Unfortunately, the theorems are too technical to state with complete precision to a non-expert, so they are frequently misinterpreted. This is also the reason it’s basically impossible to give an accurate ELI5 about these theorems, since they deal with concepts like “truth” and “provability” that have precise mathematical meanings that don’t necessarily align with popular use of the terms; hence, any truly simple description of the theorems will almost certainly gloss over crucial details and promote a misinterpretation of the theorems.
Latest Answers