Landau in his mechanics book states the following:

a frame of reference can always be chosen in which space is homogeneous and isotropic. This is called an inertial frame.

In newtonian mechanics, ground frame is almost considered inertial, but we know that in ground frame, space is not isotropic(preferred direction due to gravity). So Landau's statement actually seems wrong if we consider newtonian mechanics.

In General Relativity, ground frame is non-inertial and free fall frame is inertial in which we know that space is really isotropic - landau's statement holds true.

Even in newtonian mechanics, some inertial frames space can still be isotropic/homogeneous, but since we found even one example where it's not - i.e ground frame, then landau's statement can't be true in general for newtonian mechanics. i.e - when a statement is made, it must either be true in all cases or statement is false.

I think, Landau makes his statement in General Relativity, but nowhere, he mentions general relativity. The statement I copied is taken from his chapter of Galillean Relativity. What's your opinions on what he meant ? Should I also be thinking in terms of General relativity and not newtonian ?

Landau & Lifshitz on p.5 in their "Mechanics" book states the following:

...a frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous. This is called an inertial frame.

In Newtonian mechanics, ground frame is almost considered inertial, but we know that in ground frame, space is not isotropic  (preferred direction due to gravity). So Landau's statement actually seems wrong if we consider Newtonian mechanics.

In General Relativity, ground frame is non-inertial and free fall frame is inertial in which we know that space is really isotropic - landau's statement holds true.

Even in newtonian mechanics, some inertial frames space can still be isotropic/homogeneous, but since we found even one example where it's not - i.e ground frame, then landau's statement can't be true in general for newtonian mechanics. i.e - when a statement is made, it must either be true in all cases or statement is false.

I think, Landau makes his statement in General Relativity, but nowhere, he mentions general relativity. The statement I copied is taken from his chapter of Galillean Relativity. What's your opinions on what he meant ? Should I also be thinking in terms of General relativity and not newtonian ?

It's known that in inertial frame, space is isotropic.

When we talk about an uniform accelerated train, ground frame is considered as inertial frame. So if ground frame is considered such as inertial, then if we bring an example of dropping a ball, space should be isotropic, but we know that it's not(due to $mgy$). So I end up in a contradiction.

Where am I making a mistake ? is it that I'm wrong that ground frame can never be considered inertial or what exactly ?

Landau in his mechanics book states the following:

a frame of reference can always be chosen in which space is homogeneous and isotropic. This is called an inertial frame.

In newtonian mechanics, ground frame is almost considered inertial, but we know that in ground frame, space is not isotropic(preferred direction due to gravity). So Landau's statement actually seems wrong if we consider newtonian mechanics.

In General Relativity, ground frame is non-inertial and free fall frame is inertial in which we know that space is really isotropic - landau's statement holds true.

Even in newtonian mechanics, some inertial frames space can still be isotropic/homogeneous, but since we found even one example where it's not - i.e ground frame, then landau's statement can't be true in general for newtonian mechanics. i.e - when a statement is made, it must either be true in all cases or statement is false.

I think, Landau makes his statement in General Relativity, but nowhere, he mentions general relativity. The statement I copied is taken from his chapter of Galillean Relativity. What's your opinions on what he meant ? Should I also be thinking in terms of General relativity and not newtonian ?