The viability all depends on how easy it will be to bring PSN games to PS4 without using integrated hardware.

Having to spend ~$40 for cell integration would pay off quickly if that meant the thousands of games already on PSN available for download to PS4 customers over its lifetime, and there was no other way to achieve this. As well as the extra consumers it would incentivise to upgrade to PS4 as it will play their PS3 games.

But if, say, ~80/90% of PSN's library can be ported over to PS4 either through PC versions, Gaikai streaming or software emulation, that ~$40 upfront hit seems like a less desirable route to take. 

Well, an external chip doesn't seem viable at all. Processors can't be built as an external chip communicating with ports. Consumers would rather like to opt for a product that has out of the box feature. The external HD DVD player for 360 failed on many levels and finally took the blow with Toshiba opting out of  manufacturing discs. IBM still continues to produce one form of cell processors intended for next gen machines. And the fact that Sony has virtually made a deal with AMD for an x86 arch as per the rumours also puts them in an equally likely situation of including next-gen cell, bringing back the PS3 BC. This is perhaps another reason why PS4 might be delayed further(to find ways of reducing manufacturing costs of next gen cell) so that PS3 has an extended support until then.

Streaming or software emulation seems the ideal bet.

PS3 has PS1 backwards compatibility....yet, they still sell PS1 games in the store.

Sony could have discontinued PS1 B/C long ago, but they didn't.

Your theory/reason doesn't seem to hold up that well.

Agree, particularly if it's true they'll use an APU: the GPU core included in it could emulate SPEs using a SW layer much simpler than what a general purpose CPU would need to do the same job, and a lot more efficiently, while the parts of SW running on the PPE should be portable easily anyway, as PS3 used only one general purpose PPE core, so considering how much HW power increased in 7 years, even a simple and brute force emulation running on a modern mid-range multicore CPU should be able to reach the performances of the old native HW.
OTOH, HW BC would have been easy if Sony stuck to POWER architecture: in that case it would have just taken to add some SPEs to a normal multicore POWER 7+ CPU, possibly using the latest version developed by IBM, so that it could make sense to use them also as coprocessors for new SW (for physics engines, as DSPs, for graphics pre-processing, to run codecs, etc) and not only for BC.

The Gaikei purchase makes sense if it had anything to with the software support from streaming. I believe it's going to be the only solution alternatively and probably being limited to PS+/Premium subscribers who will have a huge library under their custody. Basically, POWER archictecture is the way to go for hardware emulation. IBM is continuing to produce next gen cells and I'm in high anticipation. If Sony can take the extra umph, probably wait until 2014 Q2 to reduce cell manufacturing costs, I believe it's worth it.

EDIT: http://n4g.com/news/1154204/ibm-may-introduce-the-new-cell-chip-for-sony-next-gen-console,

which allows me to expect the best from Sony

Which BC are we talking about? British Columbia BC or Backwards Compatibility BC?
If we're talking about the later... then you don't want to know how much time it took me trying to figure that out.

IMO, they should not. It would mean more $ spent on a larger power supply, cooling components, as well as the console's size would suffer as a result of having a larger motherboard with additional voltage regulation modules for the Cell CPU. $50 is way better spent on a faster GPU, or to keep the console $50 cheaper. Maintaining graphical leadership or competitive price against Wii U/ Xbox 720 matters a lot more for console sales to PS4's target market. High price is a far bigger deal breaker. We've already seen how when fat PS3 included BC but cost $500+ at launch, it sold poorly. It makes no sense to repeat the same mistake again especially since the economy is more fragile now globally than it was in 2006.

For example, at cost to Sony, $50 extra that would be like going from an HD7850 to an HD7950 GPU. That would be a massive increase in GPU power that people would notice over the course of 6-7 years. $50 doesn't sound like a lot but over 70 million consoles that's $3.5 billion extra. That means either Sony has to raise the price $50 to cover this additional expense (which means everyone who doesn't care for BC has to subsidize the desire of PS3 owners who do want this) OR Sony would need to absord this $3.5 billion loss. Neither options sound appealing because some gamers subsidizing the cost of BC for others sounds selfish, while Sony absorbind $3.5 billion in losses for BC sounds financially irresponsible.  I would wager the opportunity cost of losing ons ome PS4 sales to people who want BC or no buy is less than $3.5 billion in lost profits to Sony, which means it's a no brainer for Sony to not include the Cell in PS4. This actually explains them ditching the PS2 hardware in PS3 eventually, which should have been done at the start.

Also given Sony's poor financial situation, it would be better for them to sell Full HD (1080P) remastered versions of the most popular games. This way people who never owned the PS3 / never played some of those games would get a better gaming experience, while Sony would get more revenue. I think we all would agree that long-term it's better that Sony survives in this industry. For them to do so, PS4 can't be sold with huge losses like PS3 was.

When I went to university, I not only shared a dorm room with another roommate during my 1st year of studies, but I was able to fit a full desktop PC tower in there. If you can fit a PS4, how hard can it be to stack PS4 on top of PS3 vertically? You are telling us you can't buy 2-3 shelves and mount PS3/PS4 vertically on a wall like a picture frame? Really now? 

Then keep using the PS3 until there are enough games on PS4 that makes it worth purchasing. BC should never be a selling feature because next generation consoles are for next gen games, not old games. If there are not enough PS4 games that justify buying the console around launch, then there is no point wasting $400 on it until the game library expands. This has been the train of thought for gamers who grew up in the 80s and 90s on NES, SNES, Genesis, N64 but for some reason the younger generation of gamers does not understand that it's illogical to spend $400 on a console to play PS3 games when your existing PS3 does this for free. 

since the Spurs engine is already made by 3rd party companies already

http://www.leadtek.com/spursengine/press.aspx

with  4 SPE core's on that 28nm chip putting that inside a 3D stacked TSV would not be out of the realm of an idea, they may be able to put in two of those in the TSV stacked chip.

So what's the alternative then? Not buy any of the next generation consoles and play PS3 games for 6-7 years until PS5 launches, hoping PS5 will be BC with PS4 games? 

You cannot buy a stand for the TV? 

or a small shelf?

Or be creative like this:

You can't think of any possible way to mount 2 consoles? College is about learning to think outside the box, not taking no for an answer :)