Re: ZFC means?


Daryl McCullough wrote:
> Charlie-Boo says...
> >
> >Daryl McCullough wrote:
> >> Charlie-Boo says...
> >>
> >> >You don't really substitute terms for variables.
> >>
> >> Yes, you do. In first-order logic, if t is a term,
> >> and Phi is a wff, and x is any variable, then the
> >> result of replacing all free occurrences of x by t
> >> is another wff.
> >
> >Evey book talks about substitututing numbers for free variables in
> >order to represent an r.e. set, not terms.
>
> That's not correct. You can only substitute
> a *term* for a variable.

I'm talking about the wording not always mentioning that numbers might
have a different representation than decimal.

C-B


Now, in the case of arithmetic,
> there isn't that much difference, because each natural
> number has a corresponding canonical term, namely the
> *name* (or "numeral") for that natural number.
> So the distinction is not that critical for natural
> numbers. But the distinction becomes important when
> you are talking about values (such as reals, or sets,
> or points or lines, etc) that lack canonical names.
>
> You have to remember that a sentence is a finite string of
> symbols. A sentence about elephants doesn't actually contain
> any elephants, it can only contain perhaps the *word* "elephant",
> or some other word denoting one or more elephants. Similarly,
> a sentence about, say, real numbers cannot actually have any
> real numbers in it, it can only have words (strings of characters)
> that represent real numbers.
>
> You cannot substitute an object for a variable *unless* that
> object is a term---a string of symbols representing something.
>
> --
> Daryl McCullough
> Ithaca, NY

.

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