Sorry to disagree with you, Alice, but in the wild, bettas don’t normally hang out in “mud puddles”. 
http://www.aqualink.com/columns/k-betta1.html
I submit that what this is referring to is more like a small farm pond rather than a “mud puddle”. If they really did live in “mud puddles”, they would be subject to instant predation by virtually anything with enough smarts to pick up a piece of food off the buffet.
Anyway, here’s the skinny on these bettas vases that the OP is talking about. The plant used is a “spathiphyllum”, also known as a Peace Lily. The gimmick is supposed to be “Peace Lily with Fighting Fish”, get it? :rolleyes:
http://gardening.about.com/library/weekly/aa053100b.htm?once=true&
I don’t know any serious fishkeepers who would be comfortable keeping a betta in one of these, Kid.
Bettas don’t really eat plant roots, and the narrow top isn’t wide enough for proper oxygen exchange at the surface, even if they can breathe air if they have to. Also, a major consideration is the fact that bettas excrete toxic ammonia through their gills, which the plant can’t utilize as fertilizer and which gradually builds up in the water, poisoning the fish. This explains why so many of those bettas in betta bowls you see at the pet store are on the bottom of the bowl, dying. It’s not that they don’t have enough oxygen–it’s that they’re being poisoned by their own ammonia. Keeping a betta in one of these bowls or vases is IMO not kind to the fish. They’re such nice little fish, I think they deserve better.
And, Kid? Fish aren’t supposed to come to the top and gulp air. If you see any kind of fish other than a betta or a gourami doing this, it is a sign that the fish is in deep distress.
Also, FTR, plant roots do not “oxygenate the water”. What oxygenates water is gas exchange at the surface. Plant roots actually require oxygen themselves.
And sure, you can go ahead and set one of these up with a goldfish in there if you want to. Heck, you could put any kind of fish in there that appeals to you–as long as you realize that it’s strictly a temporary arrangement, exactly like a flower arrangement, and with about the same lifespan. The average feeder goldfish, in a betta vase, I’d give, oh, about a week, about the same as a vase of gladiolus. And hey, it’s cheaper, too, since feeder goldfish are running about 25 to 39 cents each at Wal-Mart and glads–hey, glads are pretty expensive. [heavy sarcasm there, in case you didn’t notice]
If you really want to start keeping fish, Kid, do some reading, browse the Web, go the pet store and look at fish, and do it right. 
(And I’m not even going to dignify the Jackass’s base prejudice against the noble carp with a response.
)