?FOL Transitive Closures
dennis.hamilton at acm.org
dennis.hamilton at acm.org
Tue Mar 14 17:44:40 EDT 2023
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
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