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harold aptroot wrote: because the same vacuous truth of "all elements are of type t" is true for all
t.
yes, but that statement only holds true if there is an element. the empty se contains none.
I'm brazilian and english (well, human languages in general) aren't my best skill, so, sorry by my english. (if you want we can speak in C# or VB.Net =p)





It is no more wrong to say that ∀x∈0:x∈T than it is to say that "all Leprechauns are Irish".
There is no claim that ∃x∈0:x∈T ("there exists an Irish leprechaun").
The negation of my claim is that ¬∀x∈0:x∈T ("not all leprechauns are Irish") which would have to be proven by showing that ∃x∈0:x∈T ("there exists a nonIrish leprechaun").
The (minority) mathematical faction of Constructivism holds that proof by negation is invalid; the obvious falsity of the negation is insufficient to proove the proposition. There seem to be more constructivists in this thread than have ever gathered before in one place.
An argument Harold already posed is relevent to address this stance:
The empty set is a proper subset of every other set. If an argument references a typed set, 0 must inherit the type. If it were allowed to not be of the referenced type, it could not have been a subset of the typed set. Harold is correct is typing 0 as exclusively prime...remembering that is can also be exclusively type as nonprime in another argument on another day!
Cheers!
[Sorry to bump so late in the game. I was deadset against Harold and Bob (almost angry with them for their obstinance!) for a full hour of parsing this thread before their arguments convinced me that my intuitive distaste for their proposition was wrong.]





Vijay Sringeri wrote: Why this unique ability for prime numbers ?
Which one exactly ?
Vijay Sringeri wrote: How is it possible that, any number can be expressed as product of prime factors ?
Proof.[^]
Vijay Sringeri wrote: What is it, which makes these prime numbers special ?
As you have pointed out, their properties can be used in a lot of algorithms. But this is the case for other "type" of numbers having other properties used in other type of algorithms. So your question is no easy to answer... It is like asking why knives are useful to cut something.
~RaGE();
I think words like 'destiny' are a way of trying to find order where none exists.  Christian Graus
Do not feed the troll !  Common proverb





Hey Rage,
That was a sharp, and precise reply. Thanks.
The link for "Euclid's lemma" was helpful, however it builds other theorems based on the fundamental fact that " Any non prime number can be expressed as product of prime numbers".
Lastly, what an explanation..
Rage wrote:
As you have pointed out, their properties can be used in a lot of algorithms. But this is the case for other "type" of numbers having other properties used in other type of algorithms. So your question is no easy to answer... It is like asking why knives are useful to cut something.
I just loved it, But sadly.. This is what I want someone to answer for me..





Vijay Sringeri wrote: I just loved it, But sadly.. This is what I want someone to answer for me..
Here you go:
Knives are useful to cut for their sharp edges
To alcohol! The cause of, and solution to, all of life's problems  Homer Simpson

Our heads are round so our thoughts can change direction  Francis Picabia





Vijay Sringeri wrote: The link for "Euclid's lemma" was helpful, however it builds other theorems based on the fundamental fact that " Any non prime number can be expressed as product of prime numbers".
There is nothing in mathematics that is not based on other proofs, assumptions and/or definitions.
And there is a proof for that lemma.






Hey, this is my standard answer to my children's questions.
~RaGE();
I think words like 'destiny' are a way of trying to find order where none exists.  Christian Graus
Do not feed the troll !  Common proverb





Thanks all, for showing keen interest in answering/trying to answer this questions.
But my question still remains unanswered
However, I just wanted to put some info here.
Prime number : Numbers > 1, and which has 1 and itself as it factor is prime nuber.
Composite number : All non prime numbers are composite numbers.
What about 1 then ?
1 is neither prime nor composite.





Wheter or not to include 1 in the list of prime numbers is debated among mathematicians. There are arguments to include it, and argument to not include it.
1 cant be written as a product of smaller primes except 1*1
However 1*N = N so you could always write any nyumber as a product of two primes if that was the case.







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