In order to have a definitive list of the syntax rules for English,
you first need to know in a definitive way what English is.
One problem is probably that you do not agree with your next door
neighbor on the English-ness of all sentences you may want to
check. I am not even considering vocabulary which varies greatly from
person to person.
If you take a diachronic view of the language (that means analyzing
the history of the language) you notice that syntactic structures
change over time. But there is no big boss saying at some point: today
we decide that adverbs will always be used in such and such way, with
all English speakers suddenly following the new rule. For one thing
there was no radio, TV or Internet to organize that.
So the evolution was progressive, the language evolving as new people
were taking the place of older ones. But at no time is there a precise
characterization of the structure of the language. No one knows
exactly was is dead, what is still used, what is coming into being.
I would think that that alone will garantee that no single list can
pretend to be a definitive description of the language.
This being said, you can always use some set of rule that will be a
good approximation of the language as you perceive it. In general,
choosing the rules is already the beginning a theory for the language
structure.
But there is only so much information about the language you can put
in the rules. The phrase structure rules you are using are, so called,
"context free". That mean they do not offer much facility for carrying
information such as agreement between parts of the sentence. It can be done, up to a point, but at
the expense of increasing considerably the number of
rules. Fundamentally, you need other mechanisms, or other techniques
than context-free rules, if only to keep to reasonable size the number
of rules of your grammar.
The rule are thus only a skeleton for the syntax of the language.
Such skeleton is always composed of a finite set of rules. However, as
I said above, it can be completed by other mechanisms (attibute,
features,...) that may often (not always) be seen as if you extended
the set of rules, possibly in an infinite way. But that is always only
a view of it, because the specifications one uses are necessarily
always expressed finitely.
Then there are languages with an internal structure of sentences that
may not even be compatible with this type of syntax specification.
See for example the question: What is an example of a syntactic structure that can't be represented by a BNF grammar?
And I am skipping semantics ("the bone ate the dog").
You will have your hands full, even with "only a simple set of rules".
But I would suggest you look at an introductory book on computational
linguistics before you start your project.
