All describable or recognizable complexity is part of the subcountable set of computable subsets of N. Higher infinities thus mostly contain fake elements about which nothing can be said, so they don’t feel any bigger.
"Fake elements," feels right to me. They're allegedly in there but we can't find any of them. It's funny that these elements comprise the majority of the "real" numbers.
I hold that the discovery of computation was as significant as the set theory paradoxes and should have produced a similar shift in practice. No one does naive set theory anymore. The same should have happened with classical mathematics but no one wanted to give up excluded middle, leading to the current situation. Computable reals are the ones that actually exist. Non-computable reals (or any other non-computable mathematical object) exist in the same way Russel’s paradoxical set exists, as a string of formal symbols.
Formal reasoning is so powerful you can pretend these things actually exist, but they don’t!
I see you are already familiar with subcountability so you know the rest.
What do you really mean exists - maybe you mean has something to do with a calculation in physics, or like we can possibly map it into some physical experience?
Doesn't that formal string of symbols exist?
Seems like allowing formal string of symbols that don't necessarily "exist" (or well useful for physics) can still lead you to something computable at the end of the day?
Like a meta version of what happens in programming - people often start with "infinite" objects eg `cycle [0,1] = [0,1,0,1...]` but then extract something finite out of it.
They don’t exist as concepts. A rational number whose square is 2 is (convenient prose for) a formal symbol describing some object. It happens that it does not describe any object. I am claiming that many objects described after the explosion of mathematics while putting calculus on a firmer foundation to resolve infinitesimals do not exist.
List functions like that need to be handled carefully to ensure termination. Summations of infinite series deal are a better example, consider adding up a geometric series. You need to add “all” the terms to get the correct result.
Of course you don’t actually add all the terms, you use algebra to determine a value.
This is really just the constructive/classical argument but I want to be specific.
You just named a function and specified a property you want it to have. However no function with this property meaningfully exists. We can manipulate the symbol just fine, but we can never look inside it because it’s not real. Classical mathematics was developed before computation was relevant and the question of decidability arose fairly late, it makes more sense to consider it an early attempt at what is now called intuitionistic mathematics. The halting problem disproved excluded middle.
This document plays at least two shell games, declaring that “homosexuality” as its own concept is recent (within 200 years) but then smoothly omitting this when discussing scripture, instead of analyzing scripture and then inserting the modern concept. No wonder it doesn’t find any condemnation of a concept it excluded from consideration!
It then does a similar trick where the authors of the New Testament are acknowledged to have poor Greek in many cases but then using specific word choice to claim they meant an extremely forced reading, relying on the previous trick a bit too.
There’s even a discussion of how nitpicking word choice is bad practice earlier in the same document!
In theory yes but very few patterns can be made like this and the cost of such machines is not favorable. Often a linking machine or a cut and sew stage is needed and this is far more hand powered than anything else.
There's also Frama-C, but having used both Frama-C and SPARK I can say I'd pick SPARK any day. Having a rich type system and not having to work with pointers makes proving a program so much easier.
Papal infallibility is not invoked that often. Here’s an example, in section 4 (wherefore…) [0]
In particular papal infallibility was not involved in the Protestants’ complaints, and the response to their complaints (Trent) was a council and again has nothing to do with papal infallibility.
The pope was also an absolute monarch at the time, but protestants didn’t care about that aspect.
I must confess that my knowledge of christian history from 300AD to 1800AD is what you can get from a few paragraphs in middle school social studies.
I was trying to allude to the indulgences with my "bullshit" but I failed.
I tend to focus more on the great awakenings and all the horrible things they led to in how they influenced America.
I thought Protestants were also at some point very against the way the Catholic faith focused on "here's what god meant" rather than letting people interpret the bible themselves? Papal infallibility is just a part of that.
This is just a polite fiction and no one is ever punished for giving illegal orders or for following them. Bloody Sunday resulted in absolutely no consequences for 1 Para. Tiger Force had illegal orders as their priority, were investigated and found guilty, and then absolutely nothing happened. The My Lai massacre resulted in a slap on the wrist for one guy.
You're wrong. There are some high profile cases where the system breaks down, but there are thousands times more situations that you've never heard about because everything gets done by the book and it never becomes a public outrage.
