SpaceX is going full steam ahead with Big Falcon Rocket (BFR).
Musk thinks that the booster part (currently called "Super Heavy") will be the easier part. Therefore, SpaceX is currently focusing on the upper stage (currently called "Starship" and before that "Big Falcon Spaceship" or "BFS").
For those losing track of Musk's repeated renaming of the project Wikipedia provides up-to date section on BFR nomenclature.
About renaming BFR to Starship, Musk's has said:
Technically, two parts: Starship is the spaceship/upper stage & Super Heavy is the rocket booster needed to escape Earth’s deep gravity well (not needed for other planets or moons)
All rockets capable of reaching Earth's orbit with useful payload are built with at least two stages. Sometimes more. This is due to the rocket equation. Dumping the dead weight of an empty first stage allows for more efficient use of energy. Super Heavy Starship (or BFR) is also built with two stages: the first stage is called "Super Heavy" and the second stage is called "Starship" (or BFS).
An interesting aspect of the Starship is that it will be able to fly without the Super Heavy. That will allow it to return from other planets and moons to the Earth. It will also be capable of single stage to orbit launch from the Earth without any useful payload. The capability is intended to be used for testing of entering planets atmospheres from super orbital velocities like Mars/Moon transfer velocities. According to Musk there are certain heating parameters that scale to the eighth power with regard to speed.
Should be done with first orbital prototype around June
The question asks:
Will SpaceX Starship (with or without a booster) reach space before 2021?
The question will resolve positively if we get information from SpaceX that any test version of Starship with or without a booster reached space before 1 January 2021. For purpose of this question reaching space will mean being at attitude of at least 80km. After reaching space, it could even explode without affecting how this question resolves. In case of failures before reaching space, several attempts are allowed.
The question will resolve negatively if no test is attempted or Starship will not reach space as defined above.
How about the chances that Starship reaches orbit in 2020?
Probability at 60% & rising rapidly due to new architecture