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I have some questions related to the foundations of mathematics and set theory. I've been studying set theory and math foundation for a while, and I see that mathematicians don't agree when comes about the basic objects that define mathematics. So, I read about the Fraenkel-Zermelo axiom system, and I find it very strange in some way. If we stop think about it, that axiom system begin with the premisse that all integers, or basic elements of counting, are sets, so, for example, x will be "xU{x}". But, if that is true, that wouldn't destroy the arithmetic? If is true that mathematics can be constructed only using one operator, what about the rest of the mathematics? We can't say for example 1+1=2 in FZ system, because there are no operations between elementary number, only operations in sets, so, what I'm trying to ask so lousily, is: If the ZF system is been considered, how can the basic arithmetic be putted on it? How about the every day 1+1, or the Cartesian graphs and all that stuff? We loose it? I know I'm about the ZF system, if he really would destroy the arithmetic that all math is based he wouldn't be considered, but I'm just asking for a clarification on my mistake. Thanks a lot.

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In a computer, all integers are just voltages in a transistor. But somehow, computers can still do arithmetic with integers! –  Hurkyl Jun 25 '13 at 4:26
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How long is "a while"? From your question, my best guess is that you mean "a few minutes". It's going to take a while longer for you to start to get a sense of it. –  dfeuer Jun 25 '13 at 4:31
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closed as not a real question by Austin Mohr, Adriano, Amzoti, Andres Caicedo, Micah Jun 25 '13 at 5:26

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1 Answer

Note that in the language of fields there is no symbol for subtraction, or division. How can we still write $5-\pi$ as a meaningful expression?

Well, the axioms of the field tell us that there exists some $x$ such that $x+\pi=0$, and we call that element $-\pi$, then we say that $x+(-y)$ will be abbreviated as $x-y$.

Similarly the axioms of $\sf ZF$ allow us to define, from a few axioms and $\in$ the operation of addition, multiplication, exponentiation, and so on. So when we write $1+1=2$ we actually write something immensely more complicated, which really states something like this:

"There exists a bijection between the disjoint union of two copies of $\{\varnothing\}$, and $\{\varnothing,\{\varnothing\}\}$."

You should have also noted that in the language of set theory we only have $\in$, and we don't have symbols for $\subseteq,\cup,\cap,\bigcup,\bigcap,\mathcal P,\setminus$ and other set operations and relations. How can we use those? (Which are much more fundamental to set theory, rather than addition and numbers.) Again the answer is that they are all definable from $\in$ and the axioms of $\sf ZF$, so we can write those symbols as an abbreviation of some formula.

The point of foundation is not to supply everyone with a single language for mathematics, but rather to ensure that there is a method to translate mathematics into a particular settings. Very much like the fact that you don't write programs with a hexeditor, but with a text editor compiler and you have a software in which you trust to turn the code you wrote textually into binary code that the computer can understand.

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It may be worth noting that the single predicate $\in$ is not actually all we have: it supplements the machinery of first-order predicate logic. As for how one goes about extending that logic to allow definitions, I've not yet studied that, but I'm under the impression that there are some subtle gotchas necessitating care. –  dfeuer Jun 25 '13 at 23:25
    
In the language of set theory we only have $\in$ and the logical symbols. You can even skip equality as a logical symbol if you want. Sure, we have conjunctions and negations, or implications, or whatever complete connectives system you want along with $\exists$ or $\forall$. But this comment really misses the point of the answer... –  Asaf Karagila Jun 25 '13 at 23:26
    
You're right, I should have attached it to the question instead. Sorry. –  dfeuer Jun 26 '13 at 0:08
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