Zero Degree Thinking

How can mathematicians consider a conflation of the infinite and zero?

The only way to do it, I think, is to consider both indeterminate, because both can be made to equal any number.

If this weren't possible, then:

If you subtract one from the infinite, then you still have the infinite.
If you subtract one from zero, then you have negative one.

Am I wrong about this? Puzzling.