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**definition of "mathematical object"**

...And "mathematical existence." Do these phrases have accepted definitions? Back when Dedekind was rejecting Cantor's transfinite ideas, could there have been a definition Cantor would refer to and say definitively "my (infinite) sets have mathematical existence because the criteria of the definition of mathematical object are satisfied"?

There are many examples of mathematical objects. All structures are mathematical objects:

http://math.chapman.edu/cgi-bin/structures?HomePage [Broken]

A proof is also a mathematical object.

Sets and categories are mathematical objects.

Any formal system is a mathematical object.

What is the common thread?

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