You Can Play the PS4 Without Day One Update
And you can pretty much use the console without ever connecting to the internet.
Sony's President of Worldwide Studios Shuhei Yoshida confirmed on Twitter that you can play your PlayStaion 4 without ever connecting to the internet. Thus, the Day One update is not required for the console to function.
— Shuhei Yoshida (@yosp) October 25, 2013
The PS4 will be released in North America on November 15th.