Extract 6.1
Context: This is the beginning of the tutorial. In CD2.1 students Jack and Andrew, says the tutor, have presented 'quite sophisticated arguments in different ways' using the Archimedean Property (ArchPr — see The Episode for two formulations of the Archimedean Property). In the following, the students and the tutor discuss its content, various formulations and proof.
The Episode:
Formulation and Meaning of the ArchPr
The tutor asks what the ArchPr says.
A1: For any number in the reals there is always a number that is bigger than that.
T1: Well, if I take any number x, then x+1 is bigger.
J1: I think you can get as close to x as you like. There's a number... greater than that but you can make it as close to x as... as you like.
T2. The tutor protests: if, for instance their meaning of 'close' means e, then he suggests taking x+e/2. What they are giving him, he says, is 'so trivial that is not worth giving it a name'.
A2: Find epsilon you choose just greater than nought and a number that exists. Say you've got a number x and you're choosing an e, there's always a number which is between x and x+e... I can't remember.
The tutor asks them to look in their notes. They say their ideas came from various books. He asks them to be more critical when trying to remember things. Jack reads from his notes 'the ArchPr for the natural numbers' which consists of two equivalent statements. The second of these usually takes the name of ArchPr, says the tutor, while writing it on the b/b (see fig.1b for the first statement):
("aÎÂ ($nÎN) (a<n)
He then asks them what the statement would say if it were false. Andrew says 'that there is an upper bound for the reals'. The tutor says he does not agree and Andrew changes to 'for the natural numbers'. The tutor then briefly talks about the non-Archimedean number-systems in which this holds. On the real line, he stresses, however large are the real numbers you are looking at, 'wherever you look there is a natural number' and 'that is the property of being archimedean'. He then asks for a proof of the ArchPr.
Proving the ArchPr
Jack suggests using contradiction and the Completeness Axiom and since, as Jack says, it is 'easier if it's on the board' he presents his suggestion in writing
J2 [while he is writing what is in fig1a]:...Nis bounded above... by a... and also Nis not empty. So, by the Completeness Axiom, em,... there exists a supremum,... of Nin the real numbers... so there is a least upper bound for the natural numbers and...?
Jack sounds hesitant. The tutor asks him what he wants to do with the supremum and Jack whispers 'this is strange' and stops. Then Andrew says: 'If you take something from it, then there's got to be something which is between that and your supremum. So if you say that alpha is the supremum, there's got to be some n which is greater than a-1'. Jack repeats the definition of the supremum and writes it on the b/b ('if the supremum equals a then n£a "nÎN. Then given any e >0, there exists neÎN such that a-e<ne') and stops. The tutor then suggests Jack follows Andrew's suggestion and puts e=1. Jack does and then spends a few seconds looking silently at the b/b. Then he says: 'However this [n1+1, see fig1a] is in the natural numbers so there is a contradiction'.
The tutor says that he agrees but also that he wishes to comment upon Jack's writing style which was 'presumably copied from the board'. Andrew points out that this is what the lecturer wrote. The tutor describes this writing style as 'b/b technique' which aims at conciseness and at avoiding long sentences. He then comments upon:
• Jack's $supNÎÂ. The tutor suggests two ways of writing: either
'N has a supremum'
or
'$aÎÂ such that n£a "nÎN and "e>0 $nÎN such that a-e<n'.
The supremum of N is something defined by the notation, he stresses, a function defined on non empty subsets of Â. Writing supN is either nonsense or a specific real number. And it doesn't make any sense to write for instance $27ÎÂ, he concludes.
• Jack's use of ne. Jack says he does not 'understand this notation'. The tutor replies that the lecturer was trying to 'produce a bit of inflection' and emphasises that 'you specify epsilon first and then you are choosing n, therefore your choice will have to depend upon epsilon. But that doesn't mean there is a function.' Andrew then asks whether 'we shouldn't use this notation with the subscript'. The tutor thinks they could but it 'would seem perverse to do so'. For specific values e=1 it is fine to write n1. The tutor then writes the second statement on the b/b (fig.1b). Jack suggests taking e=1/f and with Andrew they rephrase the property in terms of f. The tutor says he agrees but he also stresses that 'the natural thing is to define f to be 1/e' and thus 'Go from the known to the unknown'. The second statement of the ArchPr 'is just a different angle of the ArchPr for the natural numbers' says Andrew. The tutor calls the second statement a corollary from the ArchPr and points out that 'it should be called the ArchPr on the reals because it is the embedding of the natural numbers in the reals'.