Use checkboxes instead:
<input type="checkbox" name="checkbox" value="checkbox">
Just like you discovered: radiobuttons are primarily used when you want the user to select ONE option. Checkboxes should be your first choice when you want the user to select any number of options.
I beleive there is a way to work around this limitation (or feature) in the radiobutton, but IMHO, it's not worth the extra work 🙂