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What can you prove is infinite without defining it as infinite?
by postalrat 3y ago
What can you prove is infinite without defining it as infinite?
- AnimalMuppet 3y agoThe rational numbers.
- tshaddox 3y agoI was gonna say prime numbers.
- quickthrower2 3y agoGood one. You could argue they are defined as infinite though because the proof that there are infinitely many is not hard. If there are finite, multiply them all and add one. This number is prime. QED
- lovecg 3y agoOn that topic, this is a very common misconception about how that proof works. Multiplying all the primes and adding one does not have to be a prime (just try 2 x 3 x … x 13 + 1) - the point is it’s not divisible by any of the supposedly finite number of primes, so it must have some other prime divisor.
- quickthrower2 3y agoThanks I worded it wrong! You are correct.
- im3w1l 3y agoYour original proof is correct. Yes the number may not actually be prime, but given the assumption that you have finitely many primes it follows that the number you have constructed is prime, since it has no divisors. 2 x 3 x … x 13 + 1 isn't prime but that's because that's just a subset of the primes, not all of them. And the argument relies on using all of them.
- lovecg 3y agoThis is a confusing way to present this argument. We can’t say, by assumption we used up all the primes, so the new number must be a new prime - this doesn’t make sense. And you might as well say the new number has some new prime divisors (could be the number itself, could be new primes smaller than the number), which is at least slightly more correct. The actual proof is by contradiction, and goes something like this: 1) Let’s assume there’s a finite number of primes 2) Multiply them all and add one 3) Since we assumed 1), the result in 2) must have a divisor that’s one of those finite number of primes 4) But that’s not the case since we always get a remainder of one 3) and 4) is a contradiction, therefore 1 is false. At no point it follows that the number in 2) is a prime. The misconception also often causes people to assume that any first N primes + 1 is a prime itself, which is not correct. Someone will use this to make an insecure cryptographic system eventually :) Just kidding, but you see my point how it’s not just a technical detail.
- tshaddox 3y ago> We can’t say, by assumption we used up all the primes, so the new number must be a new prime - this doesn’t make sense. I'm not sure what you mean. Are you simply pointing out that it's a contradiction? Because, indeed, it is a contradiction, and that's the point. It's a proof by contradiction! The contradiction you state in step 4 is precisely equivalent to the contradiction "the new number must be a new prime." Stating "N has non-zero remainder when divided by each prime less than N" is equivalent to stating "N is prime." However this is not equivalent to the statement "any first N primary + 1 is prime" which is indeed a misconception.
- lovecg 3y ago> Stating "N has non-zero remainder when divided by each prime less than N" is equivalent to stating "N is prime." This confuses N, the bound on the largest prime, and the number (call it M) that’s the product of the all primes < N plus one. M is much larger than N. So what you’re really saying is: > Stating "M has non-zero remainder when divided by each prime less than N" is equivalent to stating "M is prime." This makes it clear where the error is, the correct statement should be: > Stating "M has non-zero remainder when divided by each prime less than N" is equivalent to stating "M is either a prime or a product A x B where A is a prime > N”
- postalrat 3y agoAre defined as infinite.
- Groxx 3y agoI define this sophistry as capable of wasting infinite time. We can accurately predict many things with these definitions of infinity. They are useful, unlike... whatever this is.
- AnimalMuppet 3y agoNo, the natural numbers are defined as infinite. The rationals are proven to be infinite (with the assumption that the naturals are).
- eru 3y agoThere are many alternative definitions of the rational numbers. I am sure you can come up with one that includes infinity in their clauses. But there are other definitions where that is a logical conclusion instead.
- Stratoscope 3y agoThere are infinitely many rational numbers, but there are even more irrational numbers. https://www.cantorsparadise.com/this-may-seem-more-irrational-than-rational-317fe1a9cbd https://www.cantorsparadise.com/this-may-seem-more-irrationa... https://archive.ph/m7qjz https://archive.ph/m7qjz
- 1970-01-01 3y agoA black hole's density at the singularity is my guess. I don't believe that's been proven however.
- quickthrower2 3y agoSo just grab a zero and stick it in a denominator!
- User23 3y agoYou can’t logically prove anything about anything unless it follows from the premises. Do you mean to suggest that all consequences of the premises are in some way part of the definition of said premises? That’s not irrational, but I don’t think it’s particularly illuminating either.
- postalrat 3y agoSuggesting that proving the universe is infinite isn't any different than proving anything else is infinite. And you shouldn't be surprised it's impossible.
- User23 3y agoWe can’t actually prove much at all about the universe beyond that it exists (cogito ergo sum). Basically everything else is talking about some model and not the actual universe. The map still isn’t the territory no matter how much we might wish it were.
- throwaway290 3y agoTime it takes to reach the end of a closed loop.
- eru 3y agoThe prime numbers, for example.