101–110 of 162 posts
Just a note from your friendly philosophy degree holder: Axioms are arbitrary. Use the axioms that are the most useful.
The question of whether or not the integer 1 is a prime doesn't make sense. The question is is it useful to define it as such and the answer is a resounding no.
One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1".
To be fair, 2 is also a very odd prime because it's even. So many theorems have to say, "for every odd prime..." https://math.stackexchange.com/questions/1177104/what-is-an-...
"Even" just means "divisible by 2"
"2 is the only prime that is divisible by 2" "3 is the only prime that is divisible by 3" "5 is the only prime that is divisible by 5"
...
"N is the only prime that is divisible by N"
Earlier quoted context omitted.
This isn't correct. While it's true that in second order logic the natural numbers admit categoricity, second order logic lacks axiomatic semantics. So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation. You can adopt Henkin semantics to give the naturals an interpretation, which…
> So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation. Can you explain what you mean here? Full semantics for second-order logic has a unique interpretation i.e. the standard natural numbers
Thus, although full second order Peano axioms are categorical, second order logic by itself never delivers a self‑contained model of the natural numbers. Any actual interpretation of the natural numbers in second order logic requires an infinite regress of background theories.
Can we declare 2 composite? Kind of annoying to have an even number in there.
2 being the only even prime isn't really anything fundamentally weird. Every prime is the only divisible-by-that-number prime. 2 has nothing unique about that. We only notice the case for 2 because our human languages happen to define divisible-by-2 as a word and concept. If our languages called divisible-by-3 "treven" or something like that, we'd think it weird that 3 was the only treven prime.
Other good nerd-sniping math questions: 0^0 = 1? Yes, it’s simpler that way. 0! = 1? Yes, it’s simpler that way. 0/0 = ∞? No, it’s undefined. 0.9999… = 1? Yes, it’s just two ways of expressing the same number. 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.
More a question of place-value representation systems than what most people are thinking of which is 1 - ε.
Earlier quoted context omitted.
What makes you think two is prime? Not everyone would agree with that statement as the artical points out.
The article states that historically Nicomachus of Gerasa didn't consider 2 a prime, in like 100 AD. Nowadays 2 is considered prime. Seems silly to question why someone is claiming 2 is prime if that is how it is defined in modern day. > What makes you think two is prime The current mathematical definition of a prime number
Earlier quoted context omitted.
What makes you think two is prime? Not everyone would agree with that statement as the artical points out.
The article states that historically Nicomachus of Gerasa didn't consider 2 a prime, in like 100 AD. Nowadays 2 is considered prime. Seems silly to question why someone is claiming 2 is prime if that is how it is defined in modern day. > What makes you think two is prime The current mathematical definition of a prime number
I will admit that for me it was being brainwashed through years of high school and university mathematics.
Earlier quoted context omitted.
I've always wondered what actually breaks if 1 is prime or conversely what defining 1 as not prime gives us. Got just far enough into my math degree before switching to CompSci to stay of of universities the rest of my life to want to know.
Some examples are in these comments, e.g. the Fundamental Theorem of Arithmetic. The Sieve of Eratosthenes is an amusing outcome, where 1 is the only prime if you take it literally. But also mentioned elsewhere in the thread: if we declared 1 to be a prime, then many (I daresay "most") of our theorems would have to change "prime number" to "prime number greater than one".
Other good nerd-sniping math questions: 0^0 = 1? Yes, it’s simpler that way. 0! = 1? Yes, it’s simpler that way. 0/0 = ∞? No, it’s undefined. 0.9999… = 1? Yes, it’s just two ways of expressing the same number. 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.
> 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be. The other ones, sure, but I'm not following this one.
Other good nerd-sniping math questions: 0^0 = 1? Yes, it’s simpler that way. 0! = 1? Yes, it’s simpler that way. 0/0 = ∞? No, it’s undefined. 0.9999… = 1? Yes, it’s just two ways of expressing the same number. 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.
> 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be. The other ones, sure, but I'm not following this one.