

Forget my last posting it is trash. I didn't know what I wrote.
Yes, I know what vacuous truth is. And I don't want to raise to question.
harold aptroot wrote: clearly an empty set contains only prime numbers
That is wrong, because:
any set X, the empty set 0 is a subset of X. This is equivalent to asserting that every element of 0 is an element of X, which is vacuously true since there are no elements of 0.
(from your link).
That doesn't mean it contains only prime numbers. That means that the empty set is part of prime numbers (as it's part of every other set), by meaning of vacuous truth.

Author of Primary ROleplaying SysTem
How do I take my coffee? Black as midnight on a moonless night.
War doesn't determine who's right. War determines who's left.





ihoecken wrote: That means that the empty set is part of prime numbers (as it's part of every other set), by meaning of vacuous truth. Ok, obviously
ihoecken wrote: That doesn't mean it contains only prime numbers. Why not?





Here is another true funny one: "The set of nonprimes does not contain only prime numbers"





harold aptroot wrote:
ihoecken wrote: That doesn't mean it contains only prime
numbers. Why not?
I'm no mathematician, but i guess it's because the empty set is also a subset of the set of all prime numbers.
The thing is: as the empty set contains no elements, you can't proof nothing about the type of it's elements.
This is one of the problems the Principia Mathematica had, that led Kurt Gödel to state that mathematica is incomplete.
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)





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.






Quote: Why this unique ability for prime numbers ?How is it possible that, any number can be expressed as product of prime factors ?
Both of these two questions could be answered by the fundamental theorem of aritmatic.
Quote: What is it, which makes these prime numbers special ?
You could read my article, and there are lots of referances there.
Finding prime numbers[^]





Thanks a lot, your article is very informative.





Vijay Sringeri wrote: Why this unique ability for prime numbers ?
It is not a unique ability for prime numbers, it is just that these numbers can not be subdivided any further. You can find the LCM and HCF using any numbers, but they will always be a combination of prime factorials, so using prime numbers is far easier.
Vijay Sringeri wrote: How is it possible that, any number can be expressed as product of prime factors ?
Essentially because a prime number can not be divided and a non prime number can be.
Any number n that is not prime has at least two divisors that are not 1 and n. These divisors are either prime or non prime. If they are non prime then by definition they follow the same rule as n. These numbers are smaller then n, so repeating this rule will always result in only prime divisors.
Vijay Sringeri wrote: What is it, which makes these prime numbers special ?
The fact that they are prime and can not be divided.





Member 2053006 wrote: Any number n that is not prime has at least two divisors that are not 1 and n.
what about 4 ? AFAIK, for has only one divisor thats not 1 or 4, and it's 2...
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)





The complete lack of any mention of Zero in this discussion has sucked out all meaning for me, and left me inside a total vacuum.
Since Zero multiplied, or divided (except of course Zero divided by Zero), by any number, natural perverted, or even fractional, will always be Zero: therefore Zero is the Prime of Primes, not to mention that Zero raised to any power remains Zero, not to mention that subtracting Zero from, or adding Zero to, any number leaves the number unchanged !
That any number divided by Zero is an infinity (whose ordinality, or Aleph, among other possible infinities: is ultimate ?) which cannot be conceptualized within linearly digital Turing/Von Neumann theoretical computational design, and must be expressed by some "placeholder" like "undefined," or "NaN," or will, on a practical level, in many circumstances crash a computer: is proof of its sacred power.
Zero is the unique singularity of the transition between positive and negative numbers, thus equivalent to the Omphalos, the stone of the navel of the geobody of the cosmos, which for the ancient Greeks was located at the shrine of the oracle at Delphi.
I propose to you that the infinite set of all possible prime numbers is contained within the infinity created by Zero divided by Zero like a tiny foot in a huge shoe: lots of wiggleroom no matter what #1 does, or does not, do.
best, Bill
"Takuan Sōhō died in Edo (presentday Tokyo) in December of 1645. At the moment before his death, Takuan painted the Chinese character 'meng' ("dream"), laid down his brush and died."





BillWoodruff wrote: That any number divided by Zero is an infinity
Normal mathematics defines that as undefined, not infinity.
A divisor that approaches an infinitely small value produces a value that approaches infinity.





I think he was talking about Lim(1/0), that is infinity.
for any other operation, 1/0 is undefined, just because no one can think what would be the result.
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)





Limit(lim) is still a convergence sequence. Which is the same as what I said.
And equating it to infinity is a misrepresentation of what it means mathematically.





yes, you are right, i couldn't remember of the correct simbol to use, so i used equals, as that was what my mathematical analysis (that's how it's called in my universsity) used.
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)





Because the "secret" factors A*B = C, with A prime and B prime, are the holy grail to break RSA, the asymmetric keys algorithm used world while to enforce security, with SSL, HTTPS, VPNs...
Ruffly speaking, in RSA itself, C is "the public key" and its factorization, A and B "the private key".
It is straightforward to find the factors for small numbers, but it happens that it is very hard to find such factors for large numbers (indeed  you need to extensively search for them).
To date it has been possible to break up to a 768 bit RSA key (C is 768 bits long) by using a cluster of many hundred servers. Larger keys (1024, 2048 bits) are still considered secure (needed hundred years of cluster computing time for one single key)  and they will, until some strong improvement will be performed in number theory.




