francis20520

"He also showed that no candidate set of axioms can ever prove its own consistency." So, does this mean that the Peano Axioms can never prove it's own consistency?? What does mean for simple arithmetic operations like addition and multiplication??? Because consistency means uniformity or reliability. So does this say that we might sometimes get different answers to 2 + 3??? What does it mean???

This is mindbogglingly complex stuff. I am assuming that there are SOME statements that CAN be proved. For example we can "prove" that 2 + 3 = 5 is true. But Godel "proved" that there are certain statements that cannot be proven to be correct or not?? I suppose that is what Godel's theorems show and where his genius lies. So, is there any statement yet discovered in mathematics that satisfies Godel's proof??? Have they found a statement that cannot be shown to be either true or false??

Actually I was told this in a Youtube comment discussion. I kind of get it that "math is self consistent". For example 2 + 2 should always give 4 under ALL circumstances. So, even in a parallel universe in the multiverse 2 + 2 will still be 4. So that is consistency from the point of view of a layman, right? But does Godel's theorems have implications in physics where we search for "knowledge"??? I.e. Does these 2 theorems put a limit to knowledge we can obtain about the universe through physics???

I like this ask this question regarding implications of Gödel's Incompleteness Theorems. That is, from what I have read one of the implications of Gödel is that either "Maths is inconsistent" or that "we will NEVER know everything". What does "maths is inconsistent" mean?? Can you give an example where maths is inconsistent?? Have they discovered such an inconsistency?? And does this mean that for example, we can never discover the smallest particle or the smallest time segment or whether the universe is finite or infinite?? What does it mean????

Just to confirm, here (https://en.wikipedia.org/wiki/Peano_axioms#Addition ) is (I think) where they show that 2 + 3 = 5, right???

Thanks for your quick responses. I don't much about axiomatic structures in Mathematics. Only things I know is what I read on Wikipedia. Just out of curiosity, is it PROVED in mathematics (like proving the Pythagoras theorem) that 2 + 2 = 4?? I.e. 0 + 0 = 0. 0 + 1 = 1, 1 + 1 = 2. Are these Axioms in mathematics??? Or is there a proof??? What is it called?

I am a layman trying to understand above theorems. This could be a stupid question. Does these theorems imply that we actually cannot prove that 2+2 = 4??? Is this one of the implications of these theorems???