The identity property is that there is a certain number that when operated with a number doesn't change it. The associative property is that you can change the grouping (i.e., change the position of the parenthesis) and still get the same answer. The commutative property is that you can exchange two numbers and still get the same answer. ![]() Using subtraction we can build negative numbers by subtracting a bigger number from a smaller giving us an answer in the set For instance we can start with the whole numbers such as 0, 1, 2, 3, etc. In mathematics there are names for many different types of numbers and you've encountered lots of these types already and some of these types contain the others. However, in this section, we will be using more sophisticated lanuage to refer to them, and take a look at each of their unique properties. Having established existence from the bottom up, we can identify an isomorphic copy of each more basic structure within each more encompassing one, and agree henceforth to use these copies (in $\Bbb C$, say) rather than the originals, for convenience.We have already talked about the different types of numbers in Chapter 1. ![]() Now, it is true that all this set-theoretic structure is unwanted baggage when our aim is to get on and do real analysis, in which we would like to have simply $\Bbb N\subset\Bbb Z\subset\Bbb Q\subset\Bbb R\subset\Bbb C$ (the last one for good measure). And $\Bbb Z$? Why, it is just a set of equivalence classes of pairs of elements of $\Bbb N$. But then what is $\Bbb Q$? The simplest way to define it is in terms of equivalence classes of pairs of elements of $\Bbb Z$. The simplest way to establish B is to construct $\Bbb R$ from $\Bbb Q$ via Dedekind cuts or equivalence classes of Cauchy sequences in $\Bbb Q$. Let us grant A (which is not exactly trivial) and look at B. Second, in order for the definition of a COF, and hence of the distinguished subset, to make sense, it must be proved that (A) any two COFs are isomorphic and (B) a COF exists. First, definition should go from the simple to the complex and from the elementary to the advanced-not the other way round. And then, as already said, you can use the Peano axioms (which in this scheme are now theorems instead of axioms) to define the real numbers and use that definition to prove that the axioms for the real numbers are true.ĭefining the natural numbers (to within isomorphism) as a distinguished subset of a complete ordered field (COF) seems mathematically slick, but is unsatisfactory in two ways. Using that definition, you can then prove that the Peano axioms are true. If you choose to start from the ZF axioms, then using them you can define the natural numbers using the Von Neumann definition. You can, instead, choose the basic axioms for set theory, also known as the Zermelo-Frankel axioms or ZF for short. ![]() Well, actually, there is, by making a different choice of axioms. Nonetheless, perhaps there is still a way to break this vicious loop? What you cannot do is to avoid choosing your axioms. You may not like that answer, but that's how things go. If instead you choose to start from the Peano axioms for the natural numbers, then using them you can define the real numbers, and using that definition you can prove that the axioms for the real numbers are true.Įither one of those provides an answer to your question. If you choose to start from the axioms for the real numbers, then using them you define the natural numbers exactly as you quoted, and using that that definition you can prove that the Peano axioms are true. As such, starting from the assumption that the reals exist reasonable, and extracting the natural numbers from the reals is fine.Īnd in mathematics, you have to start from some axioms. ![]() You have to build a foundation somewhere. In this setting, $(\mathbb$ and build up your number systems from there, you could never get to the actual results of analysis in a reasonable page count. We can define the real numbers axiomatically to be the unique (up to isomorphism) complete ordered field.
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