It’s very counter intuitive. As the other commenter suggested I was referring to Gödel and his incompleteness theorem.
Actually if the system you made up doesn’t work it would be possible to prove that it does inside that system as you can prove anything inside a system that doesn’t work.
That is why my comment is not entirely accurate it should actually be: Until you prove that if the system works you can’t prove that the system works.
Can you spot the difference in the logic here?
Until you prove that you can’t prove that the system you made up works.
If you only think about half live then yes it would be radioactive forever but in reality after a long time every atom would’ve decayed into non radioactive elements.
You can even calculate the expected time it would take for the random process of decay to terminate.
Complex topic. It would depend on the bomb in question. Some are more “dirty” than others.
This is waterfall method propaganda! It never works out this smoothly. They probably forgot important requirements like: the astronauts need to be alive on Mars.
Well shit I’ve one for professional work and one for open source / bad code practice experiments… Am I in trouble?!
Flying is still the safest mode of travel.
Because people that don’t care about privacy find this to be a nice feature.
There are gallery apps that let’s you sort by location and it’s nice if you want to search for the cool thing you saw once again.
My use of language could very well be incorrect. I am not a native English speaker anyways. That’s no reason to be so condescending.
I was just merely stating my credentials to have a basis of discussion but you do not seem to be interested in that.
I know. I’ve studied this extensively. I am specialized in formal logic and by extension set theory. I’ve worked with and help write actual research papers in this field where this is basic knowledge.
I’ve never claimed there to be a bijection between the reals and the natural numbers. Please point out what statement I made that is wrong. I would very much like to know.
Also no you do not have to accept choice for this to be true. ZF is perfectly acceptable to study various infinite sets with differing cardinality.
Edit: This is what I mean when I say that our intuition is broken. One set can be larger than the other but both be non-ending that is infinite.
Beeing larger does not mean it is more infinite!
That is until you meet analysis people that define a symbol for infinity (and it’s negation) and add it to the real numbers to close the set.
Also there are applications in computer science where ordering stuff after the first infinite ordinal is important and useful.
Yea unfortunately we do kinda calculate with infinity as a concrete thing sometimes in higher level maths…
That is the way it is often taught but actually both sets are infinite that is have no ends or in other words are not bounded.
The thing that is confusing to understand is that the question how many there are and how much there is diverges at infinity.
Our intuition (as finite beings) is broken here. Both sets are infinite but in one is more than in the other. That does not make one set more infinite than the other. You cannot be more unending than to literally have no end.
Infinity + infinity = 2 * infinity… 3 * infinity … Infinity * infinity = infinity^2 … Infinity^3 … Infinity^infinity …
If infinity is the “first” i.e. Natural infinity than those are all well defined things (Ordinals)
Someone is putting a lot of work into including/inviting you.
Yea my first respond would also totally be: “Fuck that person in particular” /s
I was just pointing out that your emotional plea, that this technology is just autocorrect is not an argument in any way.
For it to be one you need to explicitly state the implication of that fact. Yes architecturaly it is autocomplete but that does not obviously imply anything. What is it about autocomplete that barrs a system of the ability to understand?
Humans are made of meat but that does not imply they can’t speak or think.
That is not an argument. Let me demonstrate:
Humans can’t communicate. They are meat. They are not communicating. It’s literally meat.
Why are you surprised?
No you can have numbers past infinity op is wrong.
As for how to order past the first infinity it’s easy.
Of course first you have 1 < 2 < 3 < 4 < … Then you take a new number not equal to any of the others let’s call it omega. Define omega to be larger than the others. So 1 < omega, 2 < omega,…
This you can of course continue even further by introducing omega + 1 which is larger than omega and therefore larger than all natural numbers.
You can continue this even further by introducing a new number let’s call it lambda that is bigger than all omega + x where x is a natural number.
This can be continued forever i.e. an infinite amount of times.
Edit: that is meant by ordinal number as you define a unique order each step.
The problem is that the concept of cardinality and ordinality is the same in the finite case. That is numbers that tell you how many things there are can also be used to sort them.
This does not work past the first infinity. If you add omega to the natural numbers then the amount of numbers you have is still the first infinity.
But there are bigger cardinal infinities than the first one. For example the cardinality of the real numbers. I.e. There are more real numbers than natural numbers.
I think the statement “this system is consistent” is a practical statement that is unprovable in a sufficiently powerful consistent system.
Can you help me understand the tone of your text? To me it sounds kinda hostile as if what you said is some kind of gotcha.