/sci/ - Science & Math

>>9952720
>in principle all of mathematics can be derived from the axioms
false

We fall back to ZC. If that fails we fall back to Second-order Arithemtic/ZFC-P, if that fails too we fall back to Peano Arithmetic/ZFC-infinity, if that fails too we fall back to Primitive Recursive Arithmetic, if that fails too we fall back to Robinson Arithmetic and stuff interpretable in it like I-Delta-Zero arithmetic. If that's inconsistent too then we have big problems.

>>9952733
We have big fucking problems if ZFC fails, idiot.

>>9952720
Entirely depends what, where, and how ZFC is proven inconsistent. It could be something like adding, removing, or restricting a single axiom, to having to re-evaluate the very logic on which (our concept of) math is built on. Also,
>In principle all of mathematics can be derived from these axioms
do u even fkn Godel m8

>>9952727
I agree with this anon, it should say
>can be encoded using these axioms

>>9952720
Then any theory that can simulate ZFC must also be inconsistent but it's not necessarily the case that all of the theories that one can encode into ZFC are inconsistent. More importantly, if you encode another theory inside ZFC and that theory is inconsistent then ZFC must also be inconsistent.

>>9953065
You might be surprised just how much of mathematics works fine in second-order arithmetic

>>9953076
A lot of money will be spent on research on how to fix it, or: A mathematician's wet dream.
Isn't that the case for most sciences? Whenever a previous conviction is falsified, there is work to be done.
That's why most physicists probably hope String Theory is provably wrong.

>>9953076
>do u even fkn Gödel m8
FTFY

Metamathematics should not be part of math, just like metaphysics is not a part of physics. It would solve all the problems.

>>9954635

>>9953065
I'd go as far and say this is outright false.

Mathematics is the study of consequences given a logical framework and some axiomatic assumptions in said framework. ZFC is a common choice due to its relevance to the universe we exist in, but not a necessary choice in the scope of "mathematics"

It would obviously cause a great deal of commotion if you can show an inconsistency in ZFC

>>9953107
No. These axioms don't include logic. With ZFC alone you literally can't use commutativity of conjunction or modus ponens.

>>9956425
ZFC is formalized on top of classical predicate logic.