Re: Learning Logic and Set Theory

george says...

>There does not NEED to be a FORMAL definition!
>And coming from YOU of all people, this question is, I repeat,
>maddening! IF there were a formal definition, YOU would know it!
>YOU are an expert!

I am not! I'm a physics major. I only have a Masters Degree, at that.
I'm not an expert in anything. Don't accuse me of being an expert.

>> To me, it seems that for a formula to be said to define something,
>
>Oh, please. This is not a matter of opinion.

If there is no formal definition of what it means to
define something, then yes, it is a matter of opinion.

>The general linguistic behavior of the community does
>factually settle the issue.

According to the community, "countable" is perfectly
definable in the language of set theory.

>> you have to rely on an *interpretation* of the language
>
>And this is just idiotic bull***, Darryl. The WHOLE POINT is that
>first-order theorems are INDEPENDENT of the interpretation.

What relevance is that? Whether a formula defines something
or not has nothing to do with theoremhood. Perhaps you are
thinking that a formula Phi(x) defines a set S if

x in S <-> Phi(x) is provable

but by that criterion, only recursive sets are definable. That
is *not* the usual notion of "definable". The usual notion, as
I've said, is via a standard *interpretation* of a language.

>> So, in particular, a formula Phi(x) is said to
>> define "x is countable" in case
>>
>> Phi(x) is true in the interpretation
>> <-> x is a countable set
>
>If Phi can be true in some interpretations and false in others,
>then the question must arise about whether x can be countable
>in some interpretations and not countable in others.

You pick a *standard* interpretation, and then x is countable
iff Phi(x) is true in *that* interpretation. The equivalence

Phi(x) is true
<->
x is countable

only holds for the standard interpretation. For other interpretations,
the equivalence may not hold. "Countable" may not even be definable
for some interpretations.

>This is ridiculous.

>The question was whether the FORMULA ITSELF,
>whether phi, by itself, defined countability.

Without an interpretation, a formula doesn't define anything.
It is a formula, together with an interpretation that defines
a collection.

>This line was not worth pursuing.

As far as I understand (which doesn't mean a lot, since I'm
not an expert), what I've been telling you is the usual
notion of what it means for a formula to define something.

--
Daryl McCullough
Ithaca, NY

.

Quantcast