Flying with GOL airlines without a CPF

Your passport is enough to fly, you don't need to have CPF.

From Documents Required for Travel:

Foreign passengers:

Foreign nationals boarding domestic flights in Brazil will need to show one of the following documents:

• A valid passport.
• If you are a foreign national residing in Brazil you must show an RNE (National Registry of Foreigners) or a Protocolo de Renovação (RNE Renewal Procedure) issued by the Brazilian Federal Police.
• A diplomatic or consular ID.
• An original photo ID or valid passport if you are a citizen of a Mercosul member or associated country.

CPF should be needed only when buying the ticket (for tax purposes, as mentioned by Aganju). During check-in, on every website I used there was a checkbox "I'm a foreigner", "I don't have CPF" or something similar, or CPF field should be optional when you enter any nationality that is not Brazilian...

This really seems like a bug on their website. You can try to download their app or simply do the check in on the airport. I'm a foreigner living in Brazil and it happened to me three times already, I couldn't check in on the website, I went to the airport and did it without any issues.

Or, if you prefer, you can generate random CPF using this CPF generator - it returns a fake but valid number (11 zeros didn't pass because the website probably validates the number checksum).

It is possible to fly without this number. Just like year I took 5 GOL flights all without having a CPF and I even checked-in online, so it must be a recent bug or perhaps some previous data you entered that has then expect a CPF. In my experience I found the GOL call center extremely helpful and efficient (I had to deal with them because their system did not handle my foreign credit card over several attempts but no problem for check-in), so do give them a call regarding the issue.

I am a foreigner. I have tried through the web. It does not work without CPF. Even on mobile.. I also tried using free CPF generator. It does not work. Best thing to do is download the app from your phone and fill up the required fields and you can check in successfully!

I also had issues check-in in both with the app and on the website. Eventually I was successful on the website (using the chrome app on my moto e4 phone). First you have to select your country from the home screen. I am from the USA, and that was an option. If your country is not listed, there is an option for "other." After that it allowed me to check-in using my US passport number. Good luck!