Re: Learning Logic and Set Theory


Daryl McCullough wrote:
> Is there a formal definition of what it means to "define" something?

!@#$%!
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! YOU therefore may NOT ask THIS question!

For once TF is a step ahead of you; knowing dang well that formality
was not relevant, he instead asked me what "odd notion" of definability
I was using. This at least establishes that notions plural are both
extant
and relevant. However, "first-order definability" is HARDLY "odd"
among them.

> To me, it seems that for a formula to be said to define something,

Oh, please. This is not a matter of opinion. The general linguistic
behavior of the community does factually settle the issue.

> 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.

> 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. This is
ridiculous. The question was whether the FORMULA ITSELF,
whether phi, by itself, defined countability. This by definition
cannot be an interpretation-dependent question, ESPECIALLY
not in the first-order context where ALL the important stuff
(i.e. all the provable stuff) is interpretation-INdependent ANYhow.

This line was not worth pursuing.

.

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