Closed jkingdon closed 6 years ago
I'm not particularly invested in the name, and am fine with DECID ph
. But since it was my suggestion, as a justification of LEM ph
, I would argue it by analogy to AC(X) notation for parameterized AC. The idea is that LEM(ph) is a weak form of LEM relativized to a single proposition, and LEM(ph) for all ph is the actual axiom of excluded middle. Maybe it is easier to use just EM ph
, which can be read as "ph has excluded middle", which I think correctly conveys the sense in which this is like LEM but particularized to ph
.
By the way, this partially comes back to that axiom we discussed before, which can in this notation be written DECID A. x x = y
, which expresses that the space of object variables (not the universe that those variables range over, the variables themselves, generally taken to be indexed by natural numbers in the metatheory) has decidable equality. This allows for a lot of set.mm's reasoning by cases on distinctors.
Then again, I now think that it is better to assert the non-exclusive disjunction A. x x = y \/ NF_ x y
(strictly non-exclusive only when the universe has one element), because the first statement has some possibly undesirable decidability statements equivalent to DECID(the universe has one element) in the absence of dtru.
I'm moving the decidable distinctness discussion to #364
Yeah, I see how "p is decidable" can be seen as a sort of parameterized LEM just as "x has a choice function" can be seen as a parameterized Axiom of Choice. But I still prefer something which is a slightly more direct notational equivalent to "p is decidable" (that being a less unwieldy phrase than "x has a choice function" works in our favor).
I've submitted my work so far as pull request #365 and that could be merged, unless people want more discussion.
As discussed in some comments on #350 it would be good to have a notation for decidability. This can be used as an antecedent or a hypothesis (which probably will be a more graceful way to display classical results than to try to keep track of what does and does not depend on ax-3).
Decidability can also be used as a conclusion (if a formula implies the decidability of an arbitrary proposition, then that formula cannot be a theorem of intuitionistic logic).
My work in progress is here: https://github.com/metamath/set.mm/compare/develop...jkingdon:decidability-connective?expand=1 but before I make that into a pull request, I would like comments on:
Notation. One option at https://github.com/metamath/set.mm/issues/350#issuecomment-372088793 was
LEM ph
but that strikes me as confusing, since the Law of the Excluded Middle (LEM) is generally taken to be the assertion that all propositions are decidable, rather than being about decidability of a particular proposition. I also consideredDECIDABLE ph
but my current branch hasDECID ph
which I was thinking is a compromise between the explicitness ofDECIDABLE ph
and something even more concise (I'm a little cool on something very concise, given that there isn't a standard notation for this that I know of, short ofph is decidable
in words).Anything about the syntax or semantics of this definition. It seems straightforward (with the syntax being very much like
-.
and the semantics being, well, just likeph \/ -. ph
). But I won't claim to be any flavor of expert.Typesetting. Right now it typesets with small caps
DECID
next to the formula and spacing similar to¬
(soDECID x = y
is typeset similarly to¬ x = y
). Coming up with something which people find readable is relevant to whether this is a good notation.Anything else?