Re: Question about Basic Law V

Rupert wrote:
In 2nd-order logic we talk about sets.

No, we don't.
The first-order objects in 2nd-order logic are
set-LIKE but they are not called sets. They are
predicates. Some of them are named and others
are not; in the empty second-order axiom-set over
the empty signature, none of them are named, but the point
is, the GRAMMAR of 2nd-order logic APPLIES these
1st-order predicates to the 0th-order objects.
In PURE 2nd-order logic, one writes, Fx, NOT xeF,
because THERE ARE NO sets. Every 1st-order
predicate, however, is unavoidably bijectible xor
not bijectible with the predicate x=x. That makes
sets definable (AS PREDICATES, NOT as extensions
predicates).

It's not clear how you'd apply
the notion of "recursively enumerable" to them.

That's ridiculous. It's exactly clear that you simply define
a 2nd-order predicate that is true of and only of those
1st-order predicates that are recursively enumerable,
i.e., that are TM-confirmable.

If you refer to
predicates by their Goedel numbers then the notion of "recursively
enumerable" is applicable,

This statement is completely ignorant of the grammar of
2nd-order logic. In 2nd-order logic, you DON'T NEED to refer
to predicates BY anything: you just refer to them DIRECTLY.
You may need to refer to numbers OF TURING-machines but
they are predicatively describable as well.

but there is a difficulty in passing from a
predicate to the set it defines.

That is THE ONE difficulty that CAN NEVER arise
since there is never ANY NEED to pass to the set.
Anything one WAS going to say about the set, one
CAN say about the predicate ITSELF, DIRECTLY, INCLUDING
that it (or its extension) was recursively enumerable.


lugita15@xxxxxxxxx wrote:
True, this is how we understand second-order logic today.

No, False. Rupert's characterization was wrong for the
reasons I just mentioned.

Nevertheless, Frege did not have such an interpretation. Instead, he
called his second order variables "concepts," and interpreted them to
be predicates, not sets.

That is NOT a difference. THAT is a SIMILARITY to the
way we do it today.
THAT, from the original Frege, we have PRESERVED.

Instead, he considered sets, or extensions in
his phraseology, to be under the domain of first-order quantification,

THAT IS a DIFFERENCE. In 2nd-order logic nowadays,
WE JUST DON'T BOTHER WITH EXTENSIONS OR SETS AT ALL.
What you are basically seeking to do is add SOME NEW 2nd-order
axioms with the intent to REpresent the OLD Fregean framework
by MIRRORING 1st-order objects (predicates) at the 0th-order level
(precisely as you said, at the level UNDER the domain of FIRST-
order quantification). Your goal, presumably, is to allow 1st-order
predicates to Talk About OTHER 1ST-order predicates -- indirectly --
by talking -- directly -- about their 0th-order extensions.
Standardly,
normally, typically, here and now today, in 2nd-order logic, WE DON'T
DO that. Rather, WHEN we want to talk About 1st-order predicates,
WE USE 2ND-order predicates: THAT'S WHAT THEY'RE THERE FOR.

i.e. he considered extensions as objects.

Right.
As I said, WE DON'T DO that.
There's a GOOD REASON WHY we QUIT doing that.

.

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