By Euler L.

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**Extra info for A theorem of arithmetic and its proof**

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The simplest invariant is probably the number of elements, closely followed by the number of comparabilities. 13 An invariant a is called reconstructible ifffor all ordered sets P and Q we have that V p = VQ implies a(P) = a(Q). The most easily reconstructed invariant is the number of elements. It is simply one more than the number of elements of any card. The number of comparabilities is to be reconstructed in Exercise 25a. , a set of invariants so that two ordered sets with the same invariants must be isomorphic, then the reconstruction problem would be solved.

Knowing that the order properties are not destroyed when restricting ourselves to a subset, the following definition is sound. 2 Let (P, ~p) be an ordered set and let Q S; P. If Q S; P and ~Q=~P IQxQ we will call (Q, ~Q) an ordered subset (or subposet) of P. Unless indicated otherwise we will always assume that subsets of ordered sets carry the order induced by the surrounding ordered set. 3 Every set of sets S that is ordered by inclusion is an ordered subset of P (U S). Order-theoretical properties mayor may not carry over to ordered subsets.

Then A is called cofinal (coinitial) in B iff for every b E B there is an a E A such that a > b (a:::: b). 9 Let P be an ordered set. Then for every chain C a well-ordered cofinal subchain W ~ C. ~ P there is Proof. Left as Exercise 10. 7. 3 A Remark on Duality Zorn's Lemma and the Well-Ordering Theorem have an "upwards bias". Indeed, all arguments are geared towards increasing in size and then finding a bound above. Since most people would agree that an increase in the size of a set means we are "going up", this is a natural visualization.

### A theorem of arithmetic and its proof by Euler L.

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