The 'sleight' of collection

As a Math PhD student, who was first a philosophy student, I came to the conclusion that math works because of several dirty tricks. The main one was the 'sleight' (as in 'sleight of hand') where the name of the last element of a set is used as the name of the set. For example the entire set { 1, 2, 3 } is identified with the cardinal number '3'. But the one inside the braces is a label used in counting and the one for the overall set borrows that label by convention. Either way it should be hugely illegal in any logical system. I call that the sleight of collection. Mathematicians would call this an equating of ordinal numbers with cardinal ones but that ignores the type change between elements of a set and the entire set. I am confident that if you tried to keep the labels of cardinal numbers and ordinal numbers separate, math would stop working.
Same for equating the position '1' with the path of travel from 0 to that position and its quantity. Another hugely illegal move. But that seems to be what makes math work. I hate to say it but it looks like Russell's paradox actually makes things work!