# What are certain facts about our current formulation of mathematics that make us question, whether a better formulation exists?

About a year ago, I asked a question in StackExchange Physics, Can a perfectly mathematically describable universe exist in a multiverse?. It yielded some insightful answers. However, one struck a particular question in mind:

You are going to have to tell us exactly what you mean by "describable" and "predictable" so we can relate them to conventional metamathematical concepts. In mathematics there are plenty of problems that are "undecidable" but I'm not sure if that corresponds more to your definition of "un-describable" or "un-predictable". Further, it's not even known if we (modern human beings) are even working with the most complete/correct form of mathematics. Most of our math (rigorous formulations of calculus, etc..) is derived from the axioms of Zermelo-Fraenkel set theory with the axiom of choice, or ZFC for short. This formulation of set theory (while brilliant) is merely humanity's best attempt so far to formulate and formalize a theory of logic of sets. This then can house things such as the Peano Axioms which give rise to a theory of numbers and so on, but still the truth of the Axioms always remains in question. The fact that we HAVE undecidable statements in our formal systems is reason (I'll avoid the word "evidence" here) enough to suspect our formulation of mathematics may still need refining by future generations. There's always the (dreadful) possibility that homo-sapiens, as sharp as we are, may simply not have the reasoning power to accomplish the sort of "mathematical describability" needed to "describe" a universe in a multiverse.

Specifically:

The fact that we HAVE undecidable statements in our formal systems is reason (I'll avoid the word "evidence" here) enough to suspect our formulation of mathematics may still need refining by future generations

What are some other things that one might note, that make us question whether a better formulation of mathematics exists?

(PS: This is a soft question coming from a beginner. Terribly sorry if this is off-topic or badly framed, feel free to suggest, edit, or vote to close in the case it doesn't hold up to Math SE standards.)

• What do you mean by "better formulation"? Do you simply mean a different choice of axioms? – probably_someone Dec 25 '16 at 11:53
• Not quite an answer, but a counter to the idea that this should make us question our choice of axiom system based on the reason that there are undecidable statements alone: any recursively-enumerable set of axioms will have statements which are undecidable. It would seem, then, that the more important question is "Are there undecidable results which are so close to our intuition that we should include them as axioms?" I don't know if that has a satisfactory answer or if it's even well-defined, though. – Hayden Dec 25 '16 at 11:57
• The fact that we HAVE undecidable statements in our formal systems is reason enough... I seriously doubt this reasoning. Meaning, I doubt that there could ever be a formal system without undecidable statements. But nevertheless, mathematics (science in general) always need refining. – polfosol Dec 25 '16 at 12:04