> On 24 Jan., 14:46, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > >> It would be swell if you could write it in more or less set-theoretic >> terms, since, after all, you are allegedly providing a proof in ZF. >> >> Thanks much. > > A last approach to support your understanding: > Define the set of all terminating decimals 0 =< x =< 1 in ZF. > Do all that you want to do (with respect to diagonalization). > Stop as soon as you encounter a non-terminating decimal
I've no idea what you're talking about.
Let t:N -> [0, 1) be the usual list of non-terminating decimals.
d(j) = 7 if t_j(j) != 7 = 6 else
There. Done. Just as soon as I specified what d is, it is obviously non-terminating. There's no process here to stop. The variable d was undefined and then it was defined and once defined, it is clearly a non-terminating decimal. (I even hate to use temporal talk like "once defined", but hopefully my meaning is clear enough.)
I patiently await a proof in ZF that d is not non-terminating. Fun fact: the axioms of ZF don't talk about time or stopping or anything like that. So, you'll have to rework this "argument" so that it is a valid argument in ZF.
Let's start with the theorem you want to prove, shall we? What theorem is it precisely? Is it this:
Let d be defined as above. Then d is a terminating decimal.
Is that your claim? I.e.,
Let d be defined as above. Then there is an i in N such that for all j > i, d(j) = 0.
Or do you plan on proving something else?
Thanks much. Eagerly awaiting further enlightenment, etc.
-- "Being in the ring of algebraic integers is just kind of being in a weird place, but it's no different than if you are in an Elk's Lodge with weird made up rules versus just being out in regular society." -- James S. Harris, teacher