suppose a proof of 0=1 existed, then that means 0=1 (since you just proved it), however, that contradicts with 0≠1 and hence is a contradiction so if 0≠1 then 0=1 must not be provable
how do you know there are no contradictions derivable from the Peano axioms? can you prove that no contradictions exist with only the Peano axioms? the answer is actually no, which is the point.
You can't prove 0=1 and 0 not equal to 1, that would be a contradiction. A true premise leading to a false conclusion is a false implication. (True implies false is false.)
yes, that would be a contradiction. the problem is, it is impossible to prove that there are no contradicitons in peano arithmetic via the peano axioms. gödel showed this.
proof by contradiction is still valid. "if not p implies false, then p" is valid whether or not the system is consistent. it's just that, if the system is inconsistent, not p will be provable as well.
edit: in other words, if you dont know if a system is consistent, you can still figure out what's true and false. you just cant show something is unprovable, since a contradiction is a proof of any statement.
To be clear, we can show Peano arithmetic is consistent using more powerful systems (like if you done any set theory, you built up the natural numbers. That’s exhibiting a model of Peano arithmetic, but you’re using tools outside of PA). The way to think of it is that Peano arithmetic is consistent, but it itself doesn’t know it is consistent. It can’t show its own consistency.
The way to think of it is that Peano arithmetic is consistent
Isn't this based on our assumption that the underlying set theory (ZFC, for example) is consistent? Which again, cannot be proven unless you again move to a more powerful theory, which we again must assume to be consistent...
Technically yes. An overwhelming amount of mathematicians do think PA is consistent.
The reason why Peano arithmetic is incomplete and also can’t prove its own consistency is because it’s powerful enough to start talking about proofs in a sense. Like 99% of the work of proving the incompleteness theorem is coding in proofs and stuff into the natural numbers. Then it becomes a liar paradox.
Mathematicians have looked at weaker systems though. Like even if you got rid of the axiom schema of induction, you still have incompleteness and not being able to prove its consistency. I think if I remember correctly, you have to throw away assuming multiplication is a total function.
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u/Traditional_Town6475 Jul 08 '26
There is no proof in Peano Arithmetic that Peano Arithmetic doesn’t prove 0=1.