@Terry What are all the differences between this edition and the second one? Thanks!

]]>By the way I feel they are your most impressive lecture notes, displaying the broadest array of skills and originality in their combination, do you see some other set of notes you are more proud of? (I suppose you will include related posts like that on the Montgomery uncertainty principle.)

One more question: do you feel you could easily rework out all the results you have presented at least in lecture notes? I mean given the title of the notes, give the main results with proofs? Which would give you the hardest time?

Congratulations and thanks for all the work.

]]>I dont know if you know the beautiful blue Danube walz composed by Johann Strauß. The spark for the music was provided to him by a dream containing the words ” Viana be glad oh oh why why”. So dreams often contain some hidden dormant breakthrough. Stepping out of the mist into the dawn of a new beauty.

Now my dream : 1+1= 2 or 1.

1+2= 3 or 1 or 2

1+3= 4 or 1 or 2 or 3.

2+2= 4 or 1 or 2 or 3.

2+3= 5 or 1 or 2 or 3 or 4.

5+7= 12 or 1 or 2 or………………..or 11.

It has to do with groups and circles and the fact that any chess piece can be replaced by a musical instrument and any chess game can thus be turned into a musical composition of some sort, and waves combining up to form larger waves, or breaking up into many individual smaller waves.

Thanks for a great website and your time ( youtube lectures etc…)

William.

]]>For your second question, you can perform induction on a statement such as “Either is zero, or there exists exactly one number such that .”

For the third question, strictly speaking is an operation rather than a function, but it can be made into a function once one has a little bit of set theory available: see Example 3.3.2.

]]>In chapter 2 of Analysis I, the object n++ is never defined. Assuming it is a function, it is not stated until lemma 2.2.10 that every positive natural number has a unique pre-image. However, before that, when considering Definition 2.2.1 (definition of addition), the question does come to mind whether a preimage (any, not necessarily unique) can always be obtained from n++. The questions I have are:

1. Does definition of addition (2.1.1) assume that given a number not equal to zero, one can always work out the (unique) “predecessor” to it?

2. In attempting the proof of lemma 2.2.10 (Let a be a positive number. Then there exists exactly one natural number b such that b++ = a), how do I perform induction on a when, by statement of the axiom of induction, the property should be true for 0 but clearly here is true for all a except zero. I have always taken the first element available ( in this case 1) when doing induction. However, here I am trying to work from first principles, hence this question.

3. is “n++” a function? If so, it seems to have been implicitly defined through its characterisation by axioms of natural numbers. Is that a valid way of defining functions?

Many Thanks for your time,

Naveed

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