Why prime factorization ?
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I don't get any of these arguments. There is nothing in it, yes, so what? "The nothing" can, so far, still be "zero prime numbers". Or zero of anything else for that matter because the same vacuous truth of "all elements are of type t" is true for all t. The definition of the empty set doesn't say anything about that.
ihoecken wrote:
Mathematically it's not true that it contains prime numbers. The definition says: There is nothing in it.
It doesn't have to contain prime numbers, it only has to contain only prime numbers, which is the same as saying that all numbers in it are prime, which is vacuously true.
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)
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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?
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)
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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 sub-divided 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)
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harold aptroot wrote:
and thus math is broken
yes, we know math is broken since the russell paradox.
harold aptroot wrote:
but that means that an empty set can be typed (because a type is just a predicate as well - the elements of the empty set are of all types simultaneously
you really want to get on the Principia mathematica? we both know that Kurt Gödel was the only one crazy enough to ready and understand it all.
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)
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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)
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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.
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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)
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Uh, I know what the empty set is. But the statement 'set S contains only thing X' is equivalent to 'set S has no members which are not X': in this case 'the empty set contains only primes' is equivalent to 'the empty set has no non-prime members'. And since it has no members at all, that is clearly true! The empty set also only contains blue items, non-prime items, even numbers, or any other set. Empties are weird like that, a bit like zero being divisible by everything.
I don't think those statements are equivalent, hence the issue. This is somewhat tangential to my favorite mind-bender: you're in a room with an angel, a devil and two doors, one leading to heaven, the other to hell. You don't know which being is the angel or devil, nor do you know which door leads the way. The angel will always tell the truth, and the devil will always lie. You can ask one being one question in order to save your soul; what do you ask? The answer to this question is similar to the issue of asserting those two statements are equal.
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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)
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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 non-Irish 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 non-prime in another argument on another day! Cheers! [Sorry to bump so late in the game. I was dead-set 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.]