Re: The seven step-Mathematical preliminaries

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On Wed Jun  3  0:39 , Bruno Marchal <marc...@ulb.ac.be> sent:

>
>Hi Kim, Hi Marty and others,
>
>So it is perhaps time to do some math.

It is

>Obviously this is a not a course in math, but it is an explanation
>from scratch of the seven step of the universal dovetailer argument.
>It is a shortcut, and most probably we will make some digression from
>time to time, but let us try not to digress too much.
>
>Kim, you are OK with this?

I am

I have to take into account the problem you
>did have with math, and which makes this lesson a bit challenging for
>me, and I guess for you too.

Hopefully my innocence will allow me to bypass the pedantry and orthodoxies of
the field and allow a
shortcut to a high level of understanding of the UDA. Only a complete neophyte
would have the gall to
say something like that!

>
>I begin with the very useful and elementary notion of set, as
>explained in what is called "naive set theory", and which is the base
>of almost all part of math.
>
>============================================= begin
>===============================
>
>1) SET
>
>Informal definition: a set is a collection of object, called elements,
>with the idea that it, the collection or set, can be considered itself
>as an object. It is a many seen as a one, if you want. If the set is
>not to big, we can describe it exhaustively by listing the elements,
>if the set is bigger, we can describe it by some other way. Usually we
>use accolades "{", followed by the elements, separated by commas, and
>then "}", in the exhaustive description of a set.
>
>Example/exercise:
>
>1) The set of odd natural numbers which are little than 10. This is a
>well defined, and not to big set, so we can describe it exhaustively by
>{1, 3, 5, 7, 9}. In this case we say that 7 belongs to  {1, 3, 5, 7, 9}.
>Exercise 1: does the number 24 belongs to the set {1, 3, 5, 7, 9}?

No

>
>2) the set of even natural number  which are little than 13. It is {0,
>2, 4, 6, 8, 10, 12}. OK? Some people can have a difficulty which is
>not related to the notion of set, for example they can ask themselves
>if zero (0) is really an even number. We will come back to this.
>
>3) The set of odd natural numbers which are little than 100. This set
>is already too big to describe exhaustively. We will freely describe
>such a set by a quasi exhaustion like {1, 3, 5, 7, 9, 11, ... 95, 97,
>99}.
>Exercise 2: does the number 93 belongs to the set of odd natural
>numbers which are little than 100, that is: does 93 belongs to {1, 3,
>5, 7, 9, 11, ... 95, 97, 99}?

Yes

>
>4) The set of all natural numbers. This set is hard to define, yet I
>hope you agree we can describe it by the infinite quasi exhaustion by
>{0, 1, 2, 3, ...}.
>Exercise 3: does the number 666 belongs to the set of natural numbers,
>that is does 666 belongs to {0, 1, 2, 3, ...}.

Yes

>Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
>3, ...}?

No idea what square-root(2) means. When I said I was innumerate I wasn't
kidding! I could of course look
it up or ask my mathematics teacher friends but I just know your explanation
will make theirs seem trite.

>
>
>5) When a set is too big or cumbersome, mathematician like to give
>them a name. They will usually say: let S be the set {14, 345, 78}.
>Then we can say that 14 belongs to S, for example.
>Exercise 5: does 345 belongs to S?

Clearly, yes

>A set is entirely defined by its elements. Put in another way, we will
>say that two sets are equal if they have the same elements.
>Exercise 6. Let S be the set {0, 1, 45} and let M be the set described
>by {45, 0, 1}. Is it true or false that S is equal to M?

True - unless integer position within a given sequence in a set plays a role. I
will guess that it does not

>Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
>true or false that S is equal to M?

False - the commas separate each natural number

>

Done - apart from the square root question

I hope you
>don't find them too much easy, because I intend to proceed in a way
>such that all exercise will be as easy, despite we will climb toward
>very much deeper notion. Feel free to ask question, comments, etc. I

Very excited about doing this. If you can make it all as approachable as this I
am over the moon!

>
>Next: we will see some operation on sets (union, intersection), and
>the notion of subset. If all this work, I will build a latex document,
>and make it the standard reference for the seventh step for the non
>mathematician, or for the beginners in mathematics.

What a wonderful idea!

Kim

>
>Bruno
>
>
>
>http://iridia.ulb.ac.be/~marchal/
>

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