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To me it looks like attempts to axiomatize natural numbers with first order logic. You can assure there's a "zero" and an infinite chain of successors starting
by praptak 5y ago
To me it looks like attempts to axiomatize natural numbers with first order logic.
You can assure there's a "zero" and an infinite chain of successors starting from it but you cannot assure there are no cycles disconnected from the chain.
- KhoomeiK 5y agoCan you elaborate on this / provide further reading?
- praptak 5y agoWikipedia page on Peano axioms maybe. Edit/elaborate: First order rules like "no element X has successor(X)=0" and "two distinct elements have two distinct successors" assure there is a chain of successors that starts with zero and does not cycle into itself. This however does not mean that every model is isomorphic to the natural numbers because a cycle of elements separate from this chain does not falsify any of the rules(axioms). So you need an additional rule to stipulate that every element of the set is reachable from zero. To express this you have to go outside first order logic.
- Sniffnoy 5y agoThis is incorrect. You cannot assure that there are no bi-infinite chains disconnected from the starting chain, but you can assure that there no cycles disconnected from the chain. To see this, note that Peano proves that, given x and y, if there exist z and w such that x+z=y and y+w=x, then x=y. Really, introducing natural numbers -- with addition and multiplication -- is overcomplicating this. There's absolutely no multiplication involved here; you could think of this as a purely additive thing, I suppose, but really directed graphs with outdegree 1 (i.e., functions on a set) are a better fit (if you want a simple model like that).
- praptak 5y agoI meant naturals with zero and successor only.