Would it be fair to say that thus far, we have only discussed the language by which we would reason about any kinds of objects, but (other than for examples) we have not yet formally introduced any objects about which to reason? — Pfhorrest
If there was an object with this structure, it would have these properties
vs
There is an object with the structure, so it has these properties. — fdrake
As I understand it, we’re really saying “all objects with this structure have these properties”, but that’s technically true whether or not there “really” are any objects with that structure at all. All bachelors are unmarried, even if there are no bachelors. — Pfhorrest
Since the syntax and semantics are formally distinct, this highlights the possibility of a gap between syntax and semantics of a formal language; and this gap is a site of interesting questions. — fdrake
"If you interpret the the Peano axioms in the usual way, then..." — fdrake
None of the theorems are themselves necessarily true, but it’s necessarily true that they are implied by their axioms. — Pfhorrest
This also reminds me of Kripkean relative modality, where something can be be necessary inasmuch as it is true in all worlds accessible from a reference world, even if it's not true in absolutely every world. — Pfhorrest
Is it possible to write "1 + 1 = 2" entirely in terms of sets and set operations? Like, if I understand correctly, we can say that 0 = ∅ = {}, and that 1 = {∅} = {{}}, no? So 2 = {{},{{}}}, right? — Pfhorrest
"take the thing on the left and write it as its set, take the thing on the right and write it as its set, define the sum as the successor function applied to 0 as many times as the thing on the left plus the thing on the right" — fdrake
This reminds me vaguely of a philosophical or logical problem I read about once, and can't remember the resolution to at the moment. — Pfhorrest
can't remember the resolution — Pfhorrest
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