?FOL Transitive Closures

[dennis.hamilton at acm.org]

I'm missing something.  Given N as the domain of discourse, and having 0 and 
successor and the minimum axioms for Peano Arithmetic, I don't understand how

	a < S(a)
	a < b ? b < c ?a < c
doesn't establish a transitive closure using FOL (ok, with induction schema), 
and to even emphasize the potential infinity (although I think that's already 
established via S(x)),
	?a ?b [ a < b ].

So is the claim more about set-theoretic relations in FOL and an inability to 
achieve something similar given a (suitable?) relation, R ?


From: Vaughan Pratt
Sent: Saturday, March 11, 2023 23:15
Subject: Re: ?Finitism / potential infinity requires the paraconsistent logic 
NAFL

 [ . ]

The limitation I see to FOL is that it can't express transitive closure of a 
binary relation.  If it could, the following reasoning would permit defining 
the notion of a potentially infinite set entirely within FOL.

 [ . ]

Vaughan Pratt