Solve this 100=6 (100-6)a=(100-6)b
Answers
First of all, let’s look at the formal system of arithmetic everyone works implicitly with. That’s everything that follows from the Peano’s first-order axioms and, assuming those axioms are consistent and the symbols we use interpreted in a standard way, then no, you can never prove 100=6. However, a caveat here: Godel was able to show that you can’t prove that you can’t prove that 100=6, since Peano’s system of arithmetic can’t prove about itself that it is consistent, though we all know arithmetic actually is. (Actually stronger systems can prove Peano arithmetic consistent.) Just a fun fact.
Second, it’s true that from a contradiction anything can follow. So let’s look at an inconsistent set of axioms for arithmetic that I just made up right now. Let’s add “0=1” to the axioms of Peano arithmetic. Then we have the following.
Theorem: 100=6
Proof: If 100/=6, then 0=1. But one can also prove 0=/1, which is a contradiction. Since 100/=6 is impossible, then it must be the case that 100=6.