Understanding the proof of the halting problem I came across the following example that proves that the blank tape halting problem is not decidable.



I understand the proof technique, but I just don't see how the blank tape problem is shown to be undecidable in the proof.

It says $M_{w}$ starts with a blank tape but then writes $w$ onto the tape, so there is no difference to the general halting problem.

How does this imply that the blank tape halting problem is not decidable? I can't find the logical conclusion. Couldn't we just replace the blank tape halting problem with another DECIDABLE problem and argue the same way?
The argument shows that if you can solve the blank tape halting problem then you can solve the halting problem. Since you can't solve the halting problem, it follows that you can't solve the blank tape halting problem. (Here "solve" means "decide with a program that always halts".)

It's true that the same argument shows that many other problems are undecidable. But if you try to prove that a decidable problem is undecidable this way, then something will go wrong. Try it out!