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”
The Steve Conjecture still requires universal quantification—“For ALL positive integers, this process leads to 1.” That’s above pure arithmetic level.
In pure arithmetic we can only verify specific cases:
”1 is odd → 2 → 1“
”3 is odd → 4 → 2 → 1”
”5 is odd → 6 → 3 → 4 → 2 → 1″
To prove it works for ALL numbers requires stepping above arithmetic to use induction or other higher structures.
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.
Actually, your example still goes beyond arithmetic:
“No integer solutions” is a universal statement about ALL integers
Proof by contradiction is still needed
Even reframed, it requires proving properties about ALL possible p and q
In pure arithmetic we can only check specific cases: “3² ≠ 2×2²”, “4² ≠ 2×3²”, etc. Any examples using just counting and basic operations?
The proof requires:
The concept of rational numbers (not just natural numbers)
Proof by contradiction (logical structure above arithmetic)
Divisibility properties beyond basic operations
We can only use counting and basic operations (+,-,×,÷) in pure arithmetic. Any examples that stay within those bounds?
I am looking for a counter example—one that doesn’t go above the arithmetic level for both system and level of proof—can you name any?
Both examples perfectly demonstrate my point:
√2 irrationality—requires real numbers (beyond arithmetic)
Infinite primes—requires infinity (beyond arithmetic)
These seem “simple” but actually step above pure arithmetic to even state them. That’s exactly the pattern—we need to go above arithmetic level to prove things. Can you find any famous proofs that stay purely within natural numbers?
ALL solved conjectures have their proof or system above arithmetic level
NONE that stay in arithmetic are solved
While you’re correct that arithmetic operations can be derived recursively from succession, the paper’s core insight isn’t about formal derivability. Rather, it suggests some mathematical behaviors are “irreducible”—they arise directly from how properties interact rather than from deeper patterns waiting to be discovered. This may explain why certain simple-looking conjectures resist proof: we’re seeking deeper explanations when the behaviour itself IS the fundamental interaction.
The theorem prover point is interesting but misses the key distinction:
Yes, theorem provers operate at a mechanical/syntactic level
But they still encode and use higher mathematical concepts
The proofs they verify still require going above pure arithmetic
The mechanical verification is different from staying within arithmetic concepts
In pure arithmetic we can only:
Do basic operations (+,-,×,÷)
Check specific cases
Compute results
Any examples of real conjectures proven while staying at this basic level?