Re: Question about Basic Law V


lugita15@xxxxxxxxx wrote:
Rupert wrote:
lugita15@xxxxxxxxx wrote:
Rupert wrote:
Fuckwit wrote:
On 3 Sep 2006 20:41:26 -0700, "Rupert" <rupertmccallum@xxxxxxxxx> wrote:


You need to clarify further how you are going to get the system to talk
about things like predicates.

2nd order logic.


In 2nd-order logic we talk about sets. It's not clear how you'd apply
the notion of "recursively enumerable" to them. If you refer to
predicates by their Goedel numbers then the notion of "recursively
enumerable" is applicable, but there is a difficulty in passing from a
predicate to the set it defines.

True, this is how we understand second-order logic today.
Nevertheless, Frege did not have such an interpretation. Instead, he
called his second order variables "concepts," and interpreted them to
be predicates, not sets. Instead, he considered sets, or extensions in
his phraseology, to be under the domain of first-order quantification,
i.e. he considered extensions as objects. To him, two concepts F and
G, even if exactly the same objects fall under them, may still be
different concepts. For instance, the concept "being a prime number"
is different from "for all F being a prime number," in that the first
one is predicative while the second one is not.
Any further help would be greatly appreciated.
Thank You in Advance.

[...] If you spell out the details of Frege's system I may be able to
tell you in more detail how this would come out.

See:
http://plato.stanford.edu/entries/frege-logic/


Fuckwit

I have read the article which Fuckwit directed me to. The set of
expressions in the language of Frege's system such that one can derive
a contradiction from the assumption that the extension of the concept
corresponding to this expression exists is clearly a recursively
enumerable set of expressions. But that doesn't mean that the set of
concepts corresponding to such expressions is definable in Frege's
system. Frege's system talks about concepts, not the expressions. We
could devise a way of talking about Goedel codes for expressions within
Frege's system, if we assumed that certain concepts have extensions,
but while the system would thereby be able to discuss its own syntax it
would not be able to discuss its own semantics. There would be no way
of defining when a concept corresponds to an expression.
No such complications as Godel codes are needed. If you look at the
article again, you will see discussions on lamda notation. Using the
lambda notation, you may refer to the concept corresponding to an
expression within the language itself. Thus if the set of expressions
that give rise to paradoxical concepts are recursively enumerable, then
the paradoxical concepts themeselves must also be recursively
enumerable.


What does that mean? How would you express this proposition in the
language of Frege's system? How would you go about defining the
predicate which a concept satisfies if and only if the assumption that
this concept has an extension gives rise to a contradiction?

You're confusing use and mention. If we use Goedel codes, then we can
actually talk *about* the expressions and formulate propositions about
them. With what you're talking about we are just *using* the
expressions. We have no way of defining a predicate which holds of a
particular concept if and only if its defining expression has a
particular property. If you think we can do it, *how* would you do it?

By the way, I do not see why you are so confident that the set of
expressions that give rise to paradoxical concepts *are* recursively
enumerable in the first place. After this modified Basic Law V of mine
is complete, it should be a theorem of the resultant system that there
are infinitely many objects. Then the number of concepts is
uncountable. So how do you know the the set of expressions that give
rise to paradoxical concepts are recursively enumerable?


It's trivial. I can program a computer to accept an expression as
input, and then run through all possible proofs in Frege's sytem with
the proposition that the concept defined by that expression has an
extension as a new axiom. If it finds a proof of a contradiction, then
it accepts the expression, otherwise it never halts. Then the set of
expressions we're talking about is the set of expressions accepted by
this program, therefore it is recursively enumerable. But, as you point
out, the notion of "recursively enumerable" makes no sense when applied
to concepts. That's my point.

Any further help would be highly appreciated.
Thank You in Advance.

.

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