See, kids? This is what happens when you don’t use smileys.
Now my mathematical education was pretty focused on the engineering side of things; but it’s my understanding that suggesting that the set of all positive integers may be finite is … somewhat unorthodox.
If I’m recalling my math classes correctly the proof that there is an infinite number of positive integers is really simple and is a consequence of (one of) the definitions of integers, ‘0’, ‘1’ and ‘+’. As in to construct the members of the set of all integers you start with '‘0’ add ‘1’ and call that the next member of the set. You then continue adding 1 to the last member on and on forever.
No, it doesn’t. The point of the argument is that you can never account for all the possible primes. That is what infinite means.
And it is a proof. Euclid’s, to be precise. Behold the more rigorously-stated version.
I take your point, but you clearly didn’t take mine.
I said, outright and unambiguously, that there is an infinite number of them. This isn’t novel, it isn’t new. I also outright said that even in Euclid’s time, some 2,300 years ago, there was a proof showing for at least as much as the prime integers are concerned, that such is the case. How you take the reverse position that I’m somehow actually suggesting the set of all numbers, let alone merely the primes, is finite boggles my mind.
Or, is it your premise that when one offers up a supposition for consideration and then defeats the supposition that one is actually promulgating the supposition is true? For this purpose, my point was that simply asserting there is an infinite number of them and the evidence is by counting some number of them can never reach the proof of the question we want. The reason why is that either there is an infinite number of them, in which case we can never get to the end. Or, the number could be so big that no one can count up to it in a lifetime, or even say three lifetimes. For the purpose of proof, the same result will be reached even if there is a finite number of them such that the number is simply so big that it will take several people’s lifetimes to get to it. So, you can’t draw the conclusion from the simply inability of someone to count to the number, if it were finite, as proof that it’s finite. Or, in other words: failure isn’t an argument for impossibility.
Now, of course, the number is actually infinite, which as I said, has been proved for some 2,300 years.
To Skald:
No, that’s not the point. For our purposes here, it wouldn’t matter if the number is Skewes Number, or infinity. Skewes Number is definitely finite. But you would never be able to count to it. Your inability to do so would never imply that there is actually an infinity of numbers. It wouldn’t imply that there’s at least a very large finite number of them. Hence, why a proof better than a method of exhaustion is required for anything approaching infinity. Like all proofs in mathematics, the assumptions, conditions, or restrictions are established in the proof. For instance, his initial condition is to assume that the number is finite, then he set about proving by negation the original assumption. Thus showing that it’s infinite. I don’t see where the confusion is, or why you think that your statement is somehow roughly equivalent to that proof.
By your own conditions, you’ve only disproved that 7 is the largest prime. Well, groovy. So now we know that it might still be finite, but that the number doesn’t end at 7.
Ahem,
Suggesting that Skald’s informal restatement of the proof doesen’t work because you have to account for all possible numbers, and there might only be a really, really big number of them, seems to suggest that you think numbers stop at some point.
Because it’s pretty clear how to generalize his method to higher primes. He gave a specific example of the procedure outlined in his cite, using n=4. It’s not mathematically rigorous because it doesn’t explicitly lay out how to it generalizes, but it is pretty clear.
Well sure. The idea of someone other than ourselves doing the work of cleaning up this mess is pretty appealing. That’s why it’s not cleaned up yet.
Since we’re referring to a supposed timeless being why would we think “Why hasn’t God fixed it yet?” is a realistic argument.
This may possibly be the root of the disagreement with ashman165. It’s not always possible to generate a larger prime given any prime number. Your further post sort of explained that, but the opening paragraph is wrong, and it’s wrong in a way that gets to the point of what I was posting earlier.
If you could generate a new larger prime by going through an operation on a known prime, that would in fact be proof that there are infinitely many of them, but this premise is false - there’s no prime-generating mechanism like that.
The proof that I was getting at, and is one way of Euclid’s proof, is that if you assume for the sake of argument that there are finitely many primes, then there would be a way to generate a larger prime, by multiplying and adding 1 like you said. This absurdity is the disproof of the premise. However, since we now know that the number of primes isn’t finite, our prime-generation algorithm no longer works.
Ahem, do the words “but suppose” really confuse you that much? This is rather common language used in proofs. I mean, like, um, really common. It’s particularly effective when using a proof by negation to suppose the very thing you want to disprove. I mean really. It’s quite ridiculous to suggest that when supposing something for the sake of showing it isn’t true is actually a suggestion, despite the proof disproving it, that it’s true.
Curt is quite correct. My entire issue is that his proof, by its own terms, only shows that 7 isn’t the last prime number. It doesn’t do anything in the way of proving the assertion that there is an infinite number of them.
Strinka, you may say it’s clear how to do the proof correctly, and I don’t disagree. However, that isn’t what his proof did, nor is it what it attempted to do. He proved 7 isn’t the largest prime integer by the conditions he laid out. That someone else is able to come along and extend this to the inter set is of no moment. Moreover, he specifically and correctly said that it wasn’t such a proof.
Anyway, this little detour has already taken more column space than it needed to: his proof wasn’t for all prime integers by its own terms; he admits outright that it wasn’t as much. My statements are perfectly correct and the structure and grammar I used is extremely common in setting up conditions. I will devote no more time to this little tangent.
What the fuck? There are tons of ways to generate larger primes given any prime number; granted, not necessarily computationally efficient ones, but who cares?
It’s clear to me that Skald was outlining the following proof:
Theorem: For every number, there is some larger prime number (equivalently, there are infinitely many primes)
Proof: Let n be an arbitrary number. We must show that there exists a prime larger than n. Let r be 1 + the product of all the primes less than or equal to n. This must be greater than 1. Furthermore, r is clearly not divisible by any prime less than or equal to n. However, every number greater than 1 is divisible by some prime; therefore, among the numbers which divide r, there must exist some prime > n. Q.E.D.
That’s a perfectly valid proof, and I daresay it’s obvious that that was what Skald was saying; although he used the specific example of n = 7 to illustrate the steps, it was clear that he was demonstrating that all those steps could be carried out fully generally, for arbitrary starting n. I don’t understand what any of the controversy was.
Well, actually, on re-reading Skald’s post, it might be better to phrase it this way:
Theorem: For every finite set of prime numbers, there is some prime not in that set (equivalently, there are infinitely many primes)
Proof: Let F be an arbitrary finite set of primes. We must show that there exists a prime not in F. Let r be 1 + the product of all the primes in F. This must be greater than 1. Furthermore, r is clearly not divisible by any prime in F. However, every number greater than 1 is divisible by some prime; therefore, among the numbers which divide r, there must exist some prime not in F. Q.E.D.
Corollary: For every number, there is some larger prime number
Proof: For any number n, apply the above to the set of primes <= n.
Like I said, Skald illustrated this using the specific example of F = {2, 3, 5, 7}, but it was clear that he was demonstrating that all those steps could be carried out fully generally, for arbitrary starting F. It seems over-the-top pedantry to say he was not giving a valid proof.
Well, I imagine ashman165’s complaints about Skald’s proof were just pedantry regarding “But he didn’t explicitly state that we could arbitrarily replace {2, 3, 5, 7} and do everything else the exact same.” Fine; if that’s the only problem being alleged, then at least we all understand each other on that front.
But I am perplexed as to what inspired ashman165’s original comment “I’m not quite sure I agree that your argument is an actual proof of infinite numbers, but it’s a good way to give yourself plausible belief such is the case” to CurtC. I can’t see anything objectionable in the proof-outline CurtC had given, but perhaps ashman165 can explain the alleged problem in that instance.
The pointlessness of the prime number discussion is hurting me, therefore God doesn’t exist.
Right. Is suffering the big issue in doubting God or is it inanity? 
Paradise can only be created by the participates, not by God. We are learning here to discover who we are and grow spirituality. We live in a world of possibilities, it is up to us to explore, discovery, and use the beneficial, while ignoring those with no benefits. It is true there are flaws in how God is commonly understood. A lot of flaws, brought about by mistakes in interpretation of the sacred texts. Interpretations, assumptions, conclusions, and theories will never lead us to truth, only an open mind, and a humble heart can do that.
The so called sacred texts, the interpretations etc. are not of( or from) a God but humans, who decided what they wanted to pass on as sacred, just as humans decided what a God would want or not. There is nothing that is written, learned,taught, or believed that is not of human origin. It is to each his own.Humans decide what is holy and what is not.
Because some human said God said, or did something, is just that person, and what person believes in. In reality it is not a belief in A God but belief in another person’s teaching or telling that God told Him or her such a thing. Humans like to con people a lot, so what was said or told to them could well be just their own hallucinations and other people believed it was true.
This agrees with what the atheists have been saying. One way to reconcile the existence of suffering with the idea of God is for God not to be omnipotent. And if God can’t create Paradise by himself, then he clearly isn’t.
I’m sorry but that seems too much like some canned generic answer you give to someone that asks a question you don’t have the answer to. If man can do it, why can’t god? Unless he’s not omnipotent or benevolent
All information that god could have created us with at birth.
Can you truly say that your interpretation is not flawed? That is the message I seem to be getting from you, that only you, and people who believe the same as you, have “true” knowledge that isn’t flawed. That is a weighty burden to assume, and without any proof of that either. What’s to say my understanding of god isn’t correct? Obviously, he’s supposed to be benevolent and omnipotent, yet you and many others seems to want to attribute weaknesses in his power to justify the existence of suffering.
Actually it does defeat the “mysterious ways” defense - mixing omnipotence, omniscience, and omnibenevolence completely eliminates any way for non-minimized suffering to be created. We don’t need to know anything about the ways themselves; the contradiction can be shown by results alone.
However, as the POE does still fail to address the “shoot the nonbeliever in the head” defense, so it still lacks something as proofs go.
But does God have to give us the free will to engage in pointless prime discussions?
(My adviser in grad school had the world’s largest prime number for years, so I consider myself to be above this discussion. )