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In the very first chapter of the bible, God creates all the animals, then man and woman (at the same time, it seems).
In the next chapter, he creates man, then a woman out of his rib, then all the animals.
So in what order did it actually happen? And why did at least one account get the order wrong?
The second chapter does not say that man was created before animals. I presume you are referring to 2:19 which is where God brought all the animals etc. that he HAD (past tense there you may notice) created to man to name them.
Then in v21 God makes woman, which on any reading cannot be before he made man or the animals.
So I suppose my questions to you would be If you are going to read the bible can I suggest you do it a little bit more carefully? I mean if you decide for yourself what it must say without reading it properly and then try to take it apart it will no doubt be quite easy but, possibly, not very scientific

QuoteWe're talking about the inspired word of god here, describing how the universe came into existence, right?
Well not really actually. You will note that the creation of the world is essentially covered in chapter 1 (and a little bit in chapter 2). That makes it pretty clear to me that this book is not intended as some kind of scientific manual for how the universe came into existence. Should you read the rest of the Bible you will note that it concentrates mostly on the relationship between God and Man and is largely not concerned with scientific process, fashion, food, the arts or many other areas which make up the world/universe as we know it.