How is checking all Swift.String or Swift.Int values equivalent to that? There are a lot of Swift.Int values to check (2^64, on a 64-bit platform), and even more Swift.String values (every combination of characters up to that length), but simply being infeasibly slow to run on your Pentium "in a reasonable amount of time" doesn't mean it's an "impossible function to implement, and also equivalent to the halting problem". We could easily write a function to enumerate all Swift.Int values (though it might take 100 years to run).
He throws out the parenthetical remark "it’s best to think of Int as modeling the infinite set of all integers" without explanation. When you're discussing whether something is theoretically impossible or not, thinking of a finite set as equivalent to an infinite set is brushing over a pretty major distinction.