I don't know if Bruno is about to answer this in messages I haven't 
checked yet but one can visualize the square root of 2.  If you draw a 
square one meter by one meter, then the length of the diagonal is the 
square root of 2 meters.  It is approximately 1.4.  What's relevant to 
Bruno's question is that the square root of two is greater than one but 
less than two, according to the geometry of the diagonal: the diagonal 
is more than one and less than two (a picture really helps drive this 
point home).
Now since the square root of two is more than one and less than two, it 
does -not- belong to the set {0,1, 2, 3, 4, ...}.
In other words, the square root of two is not a natural number.

kimjo...@ozemail.com.au wrote:
>
> 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
>
>
>   
>> Seven exercises are enough. Are you ready to answer them.
>>     
>
>
> 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  
>> will try to adapt myself.
>>     
>
>
> 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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