Since numbers are infinite, are there technically numbers with names like "fish" or "rock" way down the line?

My cl and I were having a bit of an argument today. I say that since numbers are infinite and just keep going and going and going, there will in fact, way up there, be numbers called such things as "fish", "cat", "dog", or names such as "Jordan" or "David". I know it's a bit mind boggling as to how far out there you'd have to go, but am I right?