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I think there is also a logistics issue with this. My countries proof of vaccination was an sms with a number that you can pull up in an online database, but another country may have a simple piece of paper easily photocopied and maybe with the name changed. So possibly masses amount of time and effort to check something that the people who are vaccinated will just inconvenience and the people not vaccinated will easily wwork around
Only letting people in the country who have common sense sounds like a good idea >

But not good for the economy.