Actually, Collatz DOES require working under these constraints if staying in arithmetic. The conjecture itself needs universal quantification (“for ALL numbers...”) to even state it. In pure arithmetic we can only verify specific cases: “4 goes to 2″, “5 goes to 16 to 8 to 4 to 2”, etc.
Alright, so Collatz will be proved, and the proof will not be done by “staying in arithmetic”. Just as the proof that there do not exist numbers p and q that satisfy the equation p² = 2 q² (or equivalently, that all numbers do not satisfy it) is not done by “staying in arithmetic”. It doesn’t matter.
This shows why it DOES matter: Both Collatz and p² = 2q² require going above arithmetic to even state them (“for ALL numbers...”). That’s the key insight—we can’t even formulate these interesting conjectures within pure arithmetic, let alone prove them. In pure arithmetic we can only check specific cases:
Actually, Collatz DOES require working under these constraints if staying in arithmetic. The conjecture itself needs universal quantification (“for ALL numbers...”) to even state it. In pure arithmetic we can only verify specific cases: “4 goes to 2″, “5 goes to 16 to 8 to 4 to 2”, etc.
Alright, so Collatz will be proved, and the proof will not be done by “staying in arithmetic”. Just as the proof that there do not exist numbers p and q that satisfy the equation p² = 2 q² (or equivalently, that all numbers do not satisfy it) is not done by “staying in arithmetic”. It doesn’t matter.
This shows why it DOES matter: Both Collatz and p² = 2q² require going above arithmetic to even state them (“for ALL numbers...”). That’s the key insight—we can’t even formulate these interesting conjectures within pure arithmetic, let alone prove them. In pure arithmetic we can only check specific cases:
“4 goes to 2”
“3² ≠ 2×2²”