[FOM] RE: Real Numbers

[Victor Makarov]

...
>Insall:
>
>Now, I shall try my hand at explaining what I think can clear this up.
>My explanation will involve the use of urelements, which are objects
>in a set theory that are not themselves sets.

We do not need urelements to explain what real numbers are.

In mathematics, every mathematical concept Z exists only in the context of a 
certain mathematical theory T (let us call this theory T as the host theory 
of the concept Z). For the most of mathematicians (not set-theorists) the 
host theory for the concept "real number" is the theory of the field R. For 
set-theorists the host theory is a set theory (ZFC, ZFA, NF, ... ). So 
asking. what real numbers are we should add "in what theory". And for the 
most of mathematicians the answer is well-known:
real numbers are elements of the field R.

Victor Makarov

http://home.nyu.edu/~yvm204/vm/vm.htm

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