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I decided to learn analysis over the summer for fun, but I'm really confused by the field properties. Why is the 2 element set of 0 and 1 a field? Addition wouldn't be satisfied, because 1 + 1 = 2, which isn't in the set.

Also, if there are a set of properties that completely determine the real number system, why does that imply that there is only one? My book says something about a one to one correspondence between the reals and some other real system that preserves the functions of + and *, but I don't understand this at all.

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    For the two element field, you define $1 + 1$ to be $0$; the number 2 is not in the field.2011-07-20
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    Your second question is tautological. If you have some properties which completely determine an object, how *could* there be more than one? They would have exactly the same properties, by definition, so they *may as well* be the same (if they aren't already the same). I'm being a bit informal here, but I don't think it would be useful for me to be much more precise.2011-07-20
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    Ack. I get it now, and I feel silly. Math seems black and white when learning; you don't know it, and then you do completely and you feel stupid. Thanks!2011-07-20
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    @Zack Please ask separate question on separate pages. The above should have been two questions.2011-07-20

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The field $\mathbb{F}_2$ is indeed a field. The important thing is to remember that addition is done modulo $2$. Thus,

$$ 1 + 1 = 2 \equiv 0 \pmod{2}, $$

and $0$ is indeed in $\mathbb{F}_2$.

Furthermore, the real numbers are the unique complete totally ordered field, and they can be constructed by say, completing the rational numbers with the metric $d(x,y) = |x - y|$, where $|\cdot|$ is absolute value. If you share with us what is written in the book you are using, we may be able to help a bit more.

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    "The set R, together with the functions + and *, determine the ordering of R. It therefore follows that the decimal expansions of elements of R are compltely determined by the triple {R, +, *}. Since the addition and multiplication of decimals follow the usual rules of arithmetic, the real number system is completely dtermined by Properties I-VII, in the sense that if we have another triple {R', +', *'} satisfying these properties then there will exist a unique one-one correspondence between R and R' preserving sums and products. Thus we may speak of *the* real number system."2011-07-20
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    Is it saying that since + and * are doing the same things and satisfy the same properties, they describe the same set?2011-07-20
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    What book is this? Furthermore, echoing Zhen's sentiment: I have a feeling that you are asking the wrong question. Your question was, "If these properties completely determine the reals, how do we know there is only one such set?" I believe the question you are trying to answer is: "How do these properties [I -- VII] completely determine the real number system?" Am I correct in this thought?2011-07-20
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    The book is "Introduction to Analysis" by Rosenlicht, and it is pretty thorough in describing how the set of properties that determine the real numbers. I had a mental picture of the reals as a box with numbers in it, so I had wondered if there could be multiple boxes, but that question and image make no sense now that I think about it. For your first comment, though: are + and * different for different fields? Can you just define them to be other functions, then?2011-07-20
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    A field has two operations $\ast_1$ and $\ast_2$. When the field is the complex numbers, real numbers, or rationals, we let $\ast_1 = +$ be standard addition and $\ast_2 = \times$ standard multiplication. The fields $\mathbb{F}_p$ require $\ast_1 = +_p$, addition mod $p$, and $\ast_2 = \times_p$, multiplication mod $p$.2011-07-20
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    One important property of the reals is that every number is either zero, a non-zero square, or the negative of a non-zero square. The non-zero square numbers are therefore precisely the positive numbers. The set of positive numbers in an ordered field determines the order (to compare two elements, take the difference and see whether it is positive, zero or negative).2011-07-20