I’m not a fan of houses built on slabs -I do love a full basement. And I also like hardwood floors pretty much everywhere but the bathrooms. So, no, don’t much like your plan.

We rent a house every summer that only has a partial basement, but the other part is crawl space, not a slab. I’m sure there is a way to do what you want. Hysterical give it a lot of thought and hire an architect.

Consider lightweight concrete poured over a conventionally framed floor system.

Lightweight concrete cannot generally be ground and polished the same way as conventional weight concrete. And sometimes not polished at all. Aggregates and processes so differ among lightweight concrete manufacturers that you will have to enquire with local suppliers as to what may be possible.

So if it's the granite/travertine look you're after in your concrete floors, likely best to stick to a slab-on-grade conventional pour. Or simply use granite and/or travertine in the first place.

Anglo, most if not all basements have slabs so not sure how you can love basements and not be fan of slabs unless you mean slab as finished floor? I feel slabs as finished floor should be done more out of practicality than the look. If the build is incorporating a slab for it's structure, it can make a lot of sense to use as the finish.

Doing slabs over single family residential wood framing is not typical. While they can certainly look and perform great, expect headaches to build and high upfront costs.

We generally recommend avoiding crawlspaces altogether so choosing a frost/stem wall with slab instead of a crawlspace makes sense for those who want some concrete for the finished floor. From a building science perspective, slabs make better envelopes than crawlspaces, even if crawl is unvented.

Topography should dictate foundation design and half-basements generally make more sense for steep sites. If a basement can't drain it's subslab gravel layer to daylight via gravity, avoiding the use of pumps, then I suggest avoiding basements and sticking to frost or stemwalls.

"There will be no utilities that will need to go in the slab"

Famous last words! Maybe today you won't need utilities on that side of the house, but you never know what will happen in the future. I would not give up all the benefits of a full basement for the looks of a finished concrete floor in some rooms.

It is all depends on where you are and the frost depth and the money. Here large number of builders use combo of slab and the basement mostly with split floorplans [reduced cost]. The reason for this is uneven height - top of basement walls vs a slab: a basement stem walls should be at least 8" above the ground [I recommend min 12"], add mudsill 1.5 + add framing 11" 7/8 + mudsill gasket , plus top of stem wall is uneven vs "grade" + 4" slab height. So the basement part is at least 21" - 22" above a slab on grade. And it is next to impossible to get floors perfectly even.

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Floor slab - it is hard to walk on, almost always will crack esp if area is large due to: tensions, shrinkage, settlement, expansive soils, etc . Never understood a point of finishing it. I had a neighbour that showed me his beautiful basement finished concrete - I did not have a strength to tell him that engineering school says = will always crack. You know it is going to look awful.

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Let me echo the Mark Bischak, If you really want this stained/polished concrete look drop the floor 3" in the area and apply concrete top.

You "can" do almost anything. Putting a full basement right next to a slab will require some tricky engineering. You'll have to excavate for the full basement, pour the walls, BACKFILL and COMPACT against the poured wall, and then footers and your slab. This is not for the faint of heart.

Slab cracking will occur - but that's why you cut joints and spec enough rebar. I have 8000 sq foot of concrete slab floors in my current project. the cracking is occurring in the saw cuts. (they will be epoxy filled when I finish my floors) but we're well over a year with the floors - and we've had tractors, forklifts and other heavy equipment in both buildings, and we've Hilti'd steel i-beams to the floor - and again - cracking is happening where it was designed - and it's not visible in the floor surface.